Susie determined that the system of equations below has infinitely many solutions.
The first line goes through the points (2, -13) and (-2, -1).
The second line goes through the points (5, -22) and (-3, 2).
Is she correct?
(1 point)
Responses
No, the lines are parallel.
No, the lines are parallel.
Yes, Susie is correct.
Yes, Susie is correct.
No, there is no point of intersection.
No, there is no point of intersection.
She may or may not be correct as the lines possibly cross.
No, the lines are parallel.
To determine whether Susie is correct or not, we can find the slopes of the two lines. If the slopes are equal, the lines are parallel and therefore have no point of intersection. If the slopes are not equal, the lines will intersect at a unique point.
For the first line, we can use the slope formula:
m1 = (y2 - y1) / (x2 - x1)
Using the given points (2, -13) and (-2, -1), we have:
m1 = (-1 - (-13)) / (-2 - 2)
= 12 / (-4)
= -3
For the second line, using the points (5, -22) and (-3, 2), we have:
m2 = (2 - (-22)) / (-3 - 5)
= 24 / (-8)
= -3
Since the slopes of both lines are equal (-3), the lines are parallel. Therefore, there is no point of intersection, and Susie is correct.
No, Susie is not correct. The lines in the system are not parallel, but they have a unique point of intersection.
Put the steps in order that are used to solve the following systems of equations by substitution.
−7x−2y=−13
x−2y=11
(10 points)
1. Solve one of the equations for one variable in terms of the other variable. For example, solve the second equation for x: x = 2y + 11.
2. Substitute the expression obtained in step 1 into the other equation. Replace the variable with the corresponding expression. In this case, replace x in the first equation with 2y + 11:
-7(2y + 11) - 2y = -13.
3. Simplify and solve the resulting equation for the variable remaining, which is y. Distribute and combine like terms:
-14y - 77 - 2y = -13.
-16y - 77 = -13.
4. Solve the equation obtained in step 3 for y. This involves isolating the variable on one side of the equation. Add 77 to both sides:
-16y = 64.
Divide both sides by -16:
y = -4.
5. Substitute the value obtained in step 4 back into one of the original equations to find the other variable. Using the second equation, substitute -4 for y:
x - 2(-4) = 11.
x + 8 = 11.
6. Solve the equation obtained in step 5 for x. Subtract 8 from both sides:
x = 3.
Therefore, the system of equations −7x−2y=−13 and x−2y=11 can be solved by substitution, with the solution x = 3 and y = -4.