# INVESTIGATING SYSTEMS OF EQUATIONS

We know the way to graph a linear equation in slope-intercept form. For example, we can use the slope and y-intercept or we can find two points that satisfy the equation and connect them with a line.

A. Graph the pair of equations together :

3x - y - 2  =  0

2x + y - 3  =  0

Let us re-write the given equations in slope-intercept form.

y  =  3x - 2

(slope is 3 and y-intercept is -2)

y  =  -2x + 3

(slope is -2 and y-intercept is 3)

Based on slope and y-intercept, we can graph the given equations.

B.  Explain how to say whether (3, -3) is a solution of the equation 3x - y - 2  =  0 without using the graph.

Substitute x = 3 and y = -3 in the equation 3x - y - 2 = 0.

3(3) - (-3) - 2  =  0  ?

9 - 3 - 2  =  0  ?

9 - 5  =  0  ?

4  =  0 ----> False

Substituting (3, -3) into the equation results in a false statement, so it is not a solution.

C.  Explain how to say whether (3, -3) is a solution of the equation 2x + y - 3  =  0 without using the graph.

Substitute x = 3 and y = -3 in the equation 2x + y - 3 = 0.

2(3) + (-3) - 3  =  0  ?

6 - 3 - 3  =  0  ?

6 - 6  =  0  ?

0  =  0 ----> False

Substituting (3, -3) into the equation results in a true statement, so it is a solution.

D.  Use the graph to explain whether (3, -3) is a solution of each equation.

If (3, -3) is on the line, it is a solution. If it is not on the line, it is not a solution.

(3, -3) is  not on the line y = 3x - 2. So, (3, -3) is not a solution of y  =  3x - 2 or 3x - y - 2 = 0.

But (3, -3) is on the line y = -2x + 3, so (3, -3) is a solution of y  = -2x + 3 or 2x + y - 3 = 0.

E.  Determine if the point of intersection is a solution of both equations.

Point of intersection : (1, 1).

Because (1, 1) satisfies both the equations.

3(1) - 1 - 2  =  0

3 - 1 - 2  =  0

3 - 3  =  0

0  =  0

2x + y - 3  =  0

2(1) + 1 - 3  =  0

2 + 1 - 3  =  0

0  =  0

So, the point of intersection is the solution of both equations.

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