Section 1
Solving Systems of Linear Equations by the Graphing Method
Before you proceed further in this chapter, make your test corrections from the previous chapter.a. $A=\text{______}$ of linear equations consists of two or more linear equations.b. $A=\text{______}$to a system of linear equations is an ordered pair that is a solution to both individual equations in the system.c. Graphically, a solution to a system of linear equations in two variables is a point where the lines ______.d. A system of equations that has one or more solutions is said to be ________.e. The solution set to an inconsistent system of equations is _______.f. Two equations in a system of linear equations in two variables are said to be ______ if they represent the same line.g. Two equations in a system of linear equations in two variables are said to be ______ if they different lines.
From the graph shown, determine the solution to the system.$$\begin{array}{l}x+y=4 \\y=2 x+1\end{array}$$GRAPH CAN'T COPY.
Determine which points are solutions to the given system. (see Example 11).$$\begin{aligned}&y=8 x-5\\&y=4 x+3\\&(-1,13),(-1,1),(2,11)\end{aligned}$$
Determine which points are solutions to the given system. (see Example 11).$$\begin{aligned}&y=-\frac{1}{2} x-5\\&y=\frac{3}{4} x-10\end{aligned}$$
Determine which points are solutions to the given system. (see Example 11).$$\begin{aligned}&2 x-7 y=-30\\&y=3 x+7\\&(0,-30),\left(\frac{3}{2}, 5\right),(-1,4)\end{aligned}$$
Determine which points are solutions to the given system. (see Example 11).$$\begin{aligned}&x+2 y=4\\&y=-\frac{1}{2} x+2\\&(-2,3),(4,0),\left(3, \frac{1}{2}\right)\end{aligned}$$
Determine which points are solutions to the given system. (see Example 11).$$\begin{array}{l}x-y=6 \\4 x+3 y=-4 \\(4,-2),(6,0),(2,4)\end{array}$$
Determine which points are solutions to the given system. (see Example 11).$$\begin{aligned}&x-3 y=3\\&2 x-9 y=1\\&(0,1),(4,-1),(9,2)\end{aligned}$$
For Exercises, the graph of a system of linear equations is given.a. Identify whether the system is consistent or inconsistent.b. Identify the equations as dependent or independent.c. Identify the number of solutions to the system.$$\begin{aligned}&y=x+3\\&3 x+y=-1\end{aligned}$$GRAPH CAN'T COPY.
For Exercises, the graph of a system of linear equations is given.a. Identify whether the system is consistent or inconsistent.b. Identify the equations as dependent or independent.c. Identify the number of solutions to the system.$$\begin{aligned}&5 x-3 y=6\\&3 y=2 x+3\end{aligned}$$GRAPH CAN'T COPY.
For Exercises, the graph of a system of linear equations is given.a. Identify whether the system is consistent or inconsistent.b. Identify the equations as dependent or independent.c. Identify the number of solutions to the system.$$\begin{aligned}&2 x=y+4\\&-4 x+2 y=2\end{aligned}$$GRAPH CAN'T COPY.
For Exercises, the graph of a system of linear equations is given.a. Identify whether the system is consistent or inconsistent.b. Identify the equations as dependent or independent.c. Identify the number of solutions to the system.$$\begin{aligned}&y=-2 x-3\\&-4 x-2 y=0\end{aligned}$$GRAPH CAN'T COPY.
For Exercises, the graph of a system of linear equations is given.a. Identify whether the system is consistent or inconsistent.b. Identify the equations as dependent or independent.c. Identify the number of solutions to the system.$$\begin{aligned}&y=\frac{1}{3} x+2\\&-x+3 y=6\end{aligned}$$GRAPH CAN'T COPY.
For Exercises, the graph of a system of linear equations is given.a. Identify whether the system is consistent or inconsistent.b. Identify the equations as dependent or independent.c. Identify the number of solutions to the system.$$\begin{aligned}&y=-\frac{2}{3} x-1\\&-4 x-6 y=6\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{array}{l}2 x+y=-3 \\-x+y=3\end{array}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&4 x-3 y=12\\&3 x+4 y=-16\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&f(x)=-2 x+3\\&g(x)=5 x-4\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&h(x)=2 x+5\\&g(x)=-x+2\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&k(x)=\frac{1}{3} x-5\\&f(x)=-\frac{2}{3} x-2\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&f(x)=\frac{1}{2} x+2\\&g(x)=\frac{5}{2} x-2\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&x=4\\&y=2 x-3\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&3 x+2 y=6\\&y=-3\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&y=-2 x+3\\&-2 x=y+1\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&y=\frac{1}{3} x-2\\&x=3 y-9\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&y=\frac{2}{3} x-1\\&2 x=3 y+3\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&4 x=16-8 y\\&y=-\frac{1}{2} x+2\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&2 x=4\\&\frac{1}{2} y=-1\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&y+7=6\\&-5=2 x\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&-x+3 y=6\\&6 y=2 x+12\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&3 x=2 y-4\\&-4 y=-6 x-8\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&2 x-y=4\\&4 x+2=2 y\end{aligned}$$GRAPH CAN'T COPY.
Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.)$$\begin{aligned}&x=4 y+4\\&-2 x+8 y=-16\end{aligned}$$GRAPH CAN'T COPY.
Identify each statement as true or false.A consistent system is a system that always has a unique solution.
Identify each statement as true or false.Dependent equations form a system that has no solution.
Identify each statement as true or false.If two lines coincide, the equations are dependent.
Identify each statement as true or false.If two lines are parallel, the equations are independent.
Write a system of equations with solution set $\{(4,5)\}.$
Write a system of equations with solution set $\{(-2,6)\}.$
Find $C$ and $D$ such that the solution set to the system is $((1,3)\}.$$$\begin{aligned}C x+2 y &=11 \\-3 x+D y &=9\end{aligned}$$
Find $M$ and $N$ such that the solution set to thesystem is $\{(2,-4)\}.$$$\begin{array}{l}3 x+M y=-22 \\N x+4 y=6\end{array}$$
Use a graphing calculator to graph each linear equation on the same viewing window. Use a Trace or Intersect feature to find the point(s) of intersection.$$\begin{aligned}&y=5.62 x+15.46\\&y=-1.96 x-11.07\end{aligned}$$
Use a graphing calculator to graph each linear equation on the same viewing window. Use a Trace or Intersect feature to find the point(s) of intersection.$$\begin{aligned}&y=-2.3 x-5.48\\&y=4.62 x+26.352\end{aligned}$$
Use a graphing calculator to graph each linear equation on the same viewing window. Use a Trace or Intersect feature to find the point(s) of intersection.$$\begin{aligned}2.4 x-4.8 y &=-9.36 \\-1.8 x+5.4 y &=12.456\end{aligned}$$
Use a graphing calculator to graph each linear equation on the same viewing window. Use a Trace or Intersect feature to find the point(s) of intersection.$$\begin{aligned}36 x-90 y &=-36 \\-15.5 x-5 y &=-80.75\end{aligned}$$