INVESTIGATING SYSTEMS OF EQUATIONS
About "Investigating systems of equations"
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.
Investigating systems of equations
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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