Solve each system by graphing, if possible. If a system is...

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
y=-\frac{5}{2} x+\frac{1}{2} \\
2 x-\frac{3}{2} y=5
\end{array}\right.
$$

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more equations that involve the same set of variables. The goal is to find the values for these variables that satisfy all equations simultaneously. These systems can be solved using various methods, including substitution, elimination, and graphing.

Graphical Solution Method

The graphical solution method involves plotting each equation on a coordinate plane and identifying the point(s) at which the graphs intersect. This visual approach helps in understanding the solutions: a single intersection corresponds to a unique solution, no intersection indicates no solution (inconsistent system), and an overlapping graph signifies infinitely many solutions (dependent system).

Consistency, Inconsistency, and Dependency

A system is consistent if there is at least one set of variable values that satisfy all equations. It is inconsistent if there is no common solution, and dependent if there are infinitely many solutions because the equations represent the same line. Identifying these conditions is important for understanding the nature of a system of equations.

Intersection of Lines

The intersection point of the lines representing the equations in a system is the solution that satisfies both equations. When the intersection point has fractional coordinates, it indicates that the solution involves fractions, which require careful calculation either through graphing or algebraic methods.

Transcript

00:02 All right, so our new question is y is equal to negative 5 or 2x plus 1 half and the second equation is 2x minus 3 over 2 y is equal to 5.

00:20 All right.

00:21 So let's put these equations in the standard form, like having the whole numbers instead of fractions.

00:27 So i will expand the first equation by 2.

00:30 And the second equation also by 2 to get it off the fractions.

00:34 If you do that, we will have a 2y here is equal to negative 5x plus 2.

00:41 And for the other equation, we will have 4x minus 3y is equal to 10.

00:50 So let's find the x and way intercepts of these two, the equation 1 and the equation 2, so that we can plot the xw intercepts and draw the lines and see the intersections.

01:03 So the first one, i will start with the first equation we got for the x intercept.

01:14 If you are looking for that, for x intercept means y is equal to zero.

01:19 So when we replace the y with zero, we will have a two times zero, which is equal to zero, so if we solve for x here, we will definitely see this 2 over 5, in other words say 0 .4.

01:38 And the y intercept that we should find, this time x will be equal to 0.

01:43 So that means when we replace the x with 0 here, it will be just a 2 here.

01:49 2y is equal to 2.

01:50 So when we solve for y, we will get y is equal to 1...

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