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A set of several equations with several variables is called a _____ of equations.

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Any set of numbers that satisfies each equation of a system is called a ______ of the system.

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If a system of equations has a solution, the system is ______.

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If a system of equations has no solution, the system is ______.

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If a system of equations has only one solution, the equations of the system are ______.

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If a system of equations has infinitely many solutions, the equations of the system are ______.

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The system $\left\{\begin{array}{l}x+y=5 \\ x-y=1\end{array}\right.$ _____ (consistent, inconsistent).

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The system $\left\{\begin{array}{l}x+y=5 \\ x+y=1\end{array}\right.$ (consistent, inconsistent).

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The equations of the system $\left\{\begin{array}{l}x+y=5 \\ 2 x+2 y=10\end{array}\right.$ are ______ (dependent, independent).

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The equations of the system $\left\{\begin{array}{l}x+y=5 \\ x-y=1\end{array}\right.$ are ______ (dependent, independent).

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The pair (1,3) ______ (is, is not) a solution of the system $\left\{\begin{array}{l}x+2 y=7 \\ 2 x-y=-1\end{array}\right.$.

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The pair (1,3) ______ (is, is not) a solution of the system $\left\{\begin{array}{l}3 x+y=6 \\ x-3 y=-8\end{array}\right.$.

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Solve each system of equations by graphing.

$$\left\{\begin{array}{l}y=-3 x+5 \\x-2 y=-3\end{array}\right.$$

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Solve each system of equations by graphing.

$$\left\{\begin{array}{l}x-2 y=-3 \\3 x+y=-9\end{array}\right.$$

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Solve each system of equations by graphing.

$$\left\{\begin{array}{l}3 x+2 y=2 \\-2 x+3 y=16\end{array}\right.$$

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Solve each system of equations by graphing.

$$\left\{\begin{array}{l}x+y=0 \\5 x-2 y=14\end{array}\right.$$

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Solve each system of equations by graphing.

$$\left\{\begin{array}{l}y=-x+5 \\3 x+3 y=30\end{array}\right.$$

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Solve each system of equations by graphing.

$$\left\{\begin{array}{l}x-3 y=-3 \\2 x-6 y=12\end{array}\right.$$

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Solve each system of equations by graphing.

$$\left\{\begin{array}{l}y=-x+6 \\5 x+5 y=30\end{array}\right.$$

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Solve each system of equations by graphing.

$$\left\{\begin{array}{l}2 x-y=-3 \\8 x-4 y=-12\end{array}\right.$$

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Use a graphing calculator to approximate the solutions of each system. Give answers to the nearest tenth.

$$\left\{\begin{array}{l}y=-5.7 x+7.8 \\y=37.2-19.1 x\end{array}\right.$$

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Use a graphing calculator to approximate the solutions of each system. Give answers to the nearest tenth.

$$\left\{\begin{array}{l}y=3.4 x-1 \\y=-7.1 x+3.1\end{array}\right.$$

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$$\left\{\begin{array}{l}y=3.4 x-1 \\y=-7.1 x+3.1\end{array}\right.$$

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Use a graphing calculator to approximate the solutions of each system. Give answers to the nearest tenth.

$$\left\{\begin{array}{l}29 x+17 y=7 \\-17 x+23 y=19\end{array}\right.$$

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Use a graphing calculator to approximate the solutions of each system. Give answers to the nearest tenth.

$$\left\{\begin{array}{l}y=x-1 \\y=2 x\end{array}\right.$$

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Solve each system by substitution, if possible.

$$\left\{\begin{array}{l}y=2 x-1 \\x+y=5\end{array}\right.$$

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Solve each system by substitution, if possible.

$$\left\{\begin{array}{l}2 x+3 y=0 \\y=3 x-11\end{array}\right.$$

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Solve each system by substitution, if possible.

$$\left\{\begin{array}{l}2 x+y=3 \\y=5 x-11\end{array}\right.$$

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Solve each system by substitution, if possible.

$$\left\{\begin{array}{l}4 x+3 y=3 \\2 x-6 y=-1\end{array}\right.$$

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Solve each system by substitution, if possible.

$$\left\{\begin{array}{l}4 x+5 y=4 \8 x-15 y=3\end{array}\right.$$

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Solve each system by substitution, if possible.

$$\left\{\begin{array}{l}x+3 y=1 \\2 x+6 y=3\end{array}\right.$$

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Solve each system by substitution, if possible.

$$\left\{\begin{array}{l}x-3 y=14 \\3(x-12)=9 y\end{array}\right.$$

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Solve each system by substitution, if possible.

$$\left\{\begin{array}{l}y=3 x-6 \\x=\frac{1}{3} y+2\end{array}\right.$$

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Solve each system by substitution, if possible.

$$\left\{\begin{array}{l}3 x-y=12 \\y=3 x-12\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}5 x-3 y=12 \\2 x-3 y=3\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}2 x+3 y=8 \\-5 x+y=-3\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}x-7 y=-11 \\8 x+2 y=28\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}3 x+9 y=9 \\-x+5 y=-3\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}3(x-y)=y-9 \\5(x+y)=-15\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}2(x+y)=y+1 \\3(x+1)=y-3\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}2=\frac{1}{x+y} \\2=\frac{3}{x-y}\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}\frac{1}{x+y}=12 \\\frac{3 x}{y}=-4\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}y+2 x=5 \\0.5 y=2.5-x\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}-0.3 x+0.1 y=-0.1 \\6 x-2 y=2\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}x+2(x-y)=2 \\3(y-x)-y=5\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}3 x=4(2-y) \\3(x-2)+4 y=0\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}x+\frac{y}{3}=\frac{5}{3} \\\frac{x+y}{3}=3-x\end{array}\right.$$

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$$\left\{\begin{array}{l}x+\frac{y}{3}=\frac{5}{3} \\\frac{x+y}{3}=3-x\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}\frac{3}{2} x+\frac{1}{3} y=2 \\\frac{2}{3} x+\frac{1}{9} y=1\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}\frac{x+y}{2}+\frac{x-y}{5}=2 \\x=\frac{y}{2}+1\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}\frac{x-y}{5}+\frac{x+y}{2}=6 \\\frac{x-y}{2}-\frac{x+y}{4}=3\end{array}\right.$$

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Solve each system by the elimination method, if possible.

$$\left\{\begin{array}{l}\frac{x-2}{5}+\frac{y+3}{2}=5 \\\frac{x+3}{2}+\frac{y-2}{3}=6\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x+y+z=3 \\2 x+y+z=4 \\3 x+y-z=5\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x-y-z=0 \\x+y-z=0 \\x-y+z=2\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x-y+z=0 \\x+y+2 z=-1 \\-x-y+z=0\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}2 x+y-z=7 \\x-y+z=2 \\x+y-3 z=2\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}2 x+y=4 \\x-z=2 \\y+z=1\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}3 x+y+z=0 \\2 x-y+z=0 \\2 x+y+z=0\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x+y+z=6 \\2 x+y+3 z=17 \\x+y+2 z=11\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x+y+z=3 \\2 x+y+z=6 \\x+2 y+3 z=2\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x+y+z=3 \\x+z=2 \\2 x+2 y+2 z=3\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x+y+z=3 \\x+z=2 \\2x+y+2 z=5\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x+2 y-z=2 \\2 x-y=-1 \\3 x+y+z=1\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x+y=2 \\y+z=2 \\3 x+3 y=2\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}3 x+4 y+2 z=4 \\6 x-2 y+z=4 \\3 x-8 y-6 z=-3\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x+y=2 \\y+z=2 \\x-z=0\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}2 x-y-z=0 \\x-2 y-z=-1 \\x-y-2 z=-1\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}x+3 y-z=5 \\3 x-y+z=2 \\2 x+y=1\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}(x+y)+(y+z)+(z+x)=6 \\(x-y)+(y-z)+(z-x)=0 \\x+y+2 z=4\end{array}\right.$$

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Solve each system by any method.

$$\left\{\begin{array}{l}(x+y)+(y+z)=1 \\(x+z)+(x+z)=3 \\(x-y)-(x-z)=-1\end{array}\right.$$

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Use systems of equations to solve each problem.

If Jonathan purchases two hamburgers and four orders of french fries for $\$ 8$ and Hannah purchases three hamburgers and two orders of fries for $\$ 8,$ what is the price of each item?

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Use systems of equations to solve each problem.

Hunter purchases two tennis rackets and four cans of tennis balls for \$102.Jana purchases three tennis rackets and two cans of tennis balls for $\$ 141 .$ What is the price of each item?

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Use systems of equations to solve each problem.

A farmer raises corn and soybeans on 350 acres of land. Because of expected prices at harvest time, he thinks it would be wise to plant 100 more acres of corn than of soybeans. How many acres of each does he plant?

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Use systems of equations to solve each problem.

There is an initiation fee to join the Pine River Country Club, as well as monthly dues. The total cost after 7 months' membership will be $\$ 3025$, and after $1 \frac{1}{2}$ years, $\$ 3850$. Find both the initiation fee and the monthly dues.

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Use systems of equations to solve each problem.

A General Jackson riverboat can travel 30 kilometers downstream in 3 hours and can make the return trip in 5 hours. Find the speed of the boat in still water.

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Use systems of equations to solve each problem.

A rectangular picture frame has a perimeter of 1900 centimeters and a width that is 250 centimeters less than its length. Find the area of the picture.

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Use systems of equations to solve each problem.

A metallurgist wants to make 60 grams of an alloy that is to be $34 \%$ copper. She that is 250 centimeters less than its length. Find the area of the picture.

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Use systems of equations to solve each problem.

The two weights shown will be in balance if the product of one weight and its distance from the fulcrum is equal to the product of the other weight and its distance from the fulcrum. Two weights are in balance when one is 2 meters and the other 3 meters from the fulcrum. If the fulcrum remained in the same spot and the weights were interchanged, the closer weight would need to be increased by 5 pounds to maintain balance. Find the weights.

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Use systems of equations to solve each problem.

A 112 -pound force can lift the 448 -pound load shown. If the fulcrum is moved 1 additional foot away from the load, a 192 -pound force is required. Find the length of the lever.

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Use systems of equations to solve each problem.

For a test question, a mathematics teacher wants to find two constants $a$ and $b$ such that the test item "Simplify $a(x+2 y)-b(2 x-y)^{\prime \prime}$ will have an answer of $-3 x+9 y .$ What constants $a$ and $b$ should the teacher use?

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Use systems of equations to solve each problem.

Rollowheel, Inc., can manufacture a pair of in-line skates for $\$ 43.53 .$ Daily fixed costs of manufacturing in-line skates amount to $\$ 742.72 .$ A pair of in-line skates can be sold for 889.95. Find equations expressing the expenses $E$ and the revenue $R$ as functions of $x$, the number of pairs manufactured and sold. At what production level will expenses equal revenues?

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Use systems of equations to solve each problem.

For its sales staff, a company offers two salary options. One is $\$ 326$ per week plus a commission of $3 \frac{1}{2} \%$ of sales. The other is $\$ 200$ per week plus $4 \frac{1}{4} \%$ of sales. Find equations that express incomes $I_{1}$ and $I_{2}$ as functions of sales $x$, and find the weekly sales level that produces equal salaries.

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Use systems of three equations in three variables to solve each problem.

A college student earns $\$ 198.50$ per week working three part-time jobs. Half of his 30-hour work week is spent cooking hamburgers at a fast-food chain, earning $\$ 5.70$ per hour. In addition, the student earns $\$ 6.30$ per hour working at a gas station and $\$ 10$ per hour doing janitorial work. How many hours per week does the student work at each job?

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Use systems of three equations in three variables to solve each problem.

A woman invested a $\$ 22,000$ rollover IRA account in three banks paying $5 \%$ $6 \%,$ and $7 \%$ annual interest. She invested $\$ 2000$ more at $6 \%$ than at $5 \% .$ The total annual interest she earned was $\$ 1370 .$ How much did she invest at each rate?

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Use systems of three equations in three variables to solve each problem.

Approximately 3 million people live in costa Rica. 2.61 million are younger than 50 years, and 1.95 million are older than 14 years. How many people are in each of the categories 0 14 years, 15- 49 years, and 50 years and older?

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Use systems of three equations in three variables to solve each problem.

The engineer designing a parabolic arch knows that its equation has the form $y=a x^{2}+b x+c .$ Use the information in the illustration to find $a, b,$ and $c$ Assume that the distances are given in feet. (Hint: The coordinates of points on the parabola satisfy its equation.)

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Use systems of three equations in three variables to solve each problem.

The sum of the angles of a triangle is $180^{\circ} .$ In a certain triangle, the largest angle is $20^{\circ}$ greater than the sum of the other two and is $10^{\circ}$ greater than 3 times the smallest. How large is each angle?

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Use systems of three equations in three variables to solve each problem.

The path of a thrown object is a parabola with the equation $f(x)=a x^{2}+b x+c$ Use the information in the illustration to find $a, b,$ and $c .$ (Distances are in feet.)

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If no method is stated, describe how you would determine the most efficient method to use to solve a linear system.

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Describe how a system of three equations in three variables can be reduced to a system of two equations and two variables.

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When using the elimination method, how can you tell whether the system has no solution?

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When using the elimination method, how can you tell whether the system has infinitely many solutions?

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Use a graphing calculator to attempt to find the solution of the system $\left\{\begin{array}{l}x-8 y=-51 \\ 3 x-25 y=-160\end{array}\right.$.

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Solve the system of Exercise 93 algebraically. Which method is easier, and why?

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Use a graphing calculator to attempt to find the solution of the system $\left\{\begin{array}{l}17 x-23 y=-76 \\ 29 x+19 y=-278\end{array}\right.$

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Solve the system of Exercise 95 algebraically. Which method is easier, and why?

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Write a system of two equations in two variables with the solution (-2,5).

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Write a system of three equations in three variables with the solution $(-4,5,1)$.

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Write a system of three equations in three variables with an infinite number of solutions.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

If a system of two equations in two variables is represented by two lines with the same slope and different $y$ -intercepts, then the system has an infinite number of solutions.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

If a system of two equations in two variables is represented by two lines with negative reciprocal slopes, then the system has an infinite number of solutions.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

If a linear system of three equations in three variables has infinitely many solutions, then any ordered triple is a solution of the system.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

When using the graphing method, a system of two equations in two variables can appear to have no solution and yet have a unique one.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

A linear system of two equations in three variables cannot have a unique solution.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

If a linear system of two equations in two variables has a solution set involving fractions, then use the graphing method to ensure accuracy.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

To solve a linear system of three equations in three variables, we use the graphing method.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

The system of equations $999 x-999 y=999$ and $-999 x+999 y=-999$ has an infinite number of =solutions.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

The system of equations $-777 x+777 y=-777$ and $777 x-777 y=-777$ has no solution.

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Determine if the statement is true or false. If the statement is false, then correct it and make it true.

The system of equations $555 x+555 y=555$ and $555 x-555 y=-555$ has no solution.

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