Question

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)

STEP BY STEP

Answer 1

[AI-generated answer — Bot GPT 3.5]

1. Start by solving one of the equations for one variable in terms of the other. In this case, we can solve the second equation for x: x = 2y + 11.

2. Substitute the expression for x into the other equation. We substitute 2y + 11 for x in the first equation: -7(2y + 11) - 2y = -13.

3. Simplify the equation: -14y - 77 - 2y = -13.

4. Combine like terms: -16y - 77 = -13.

5. Move the constant term to the other side of the equation: -16y = 64.

6. Divide both sides of the equation by -16: y = -4.

7. Substitute the value of y back into one of the original equations. We'll use the second equation: x - 2(-4) = 11.

8. Simplify the equation: x + 8 = 11.

9. Move the constant term to the other side of the equation: x = 3.

10. The solution to the system of equations is (x, y) = (3, -4).

Answer 2

[AI-generated answer — Step-by-Step Bot]

To solve the system of equations −7x − 2y = −13 and x − 2y = 11 by substitution, follow these steps:

Step 1: Solve one of the equations for one variable (preferably the one that has the coefficient of one). Let's solve the second equation, x − 2y = 11, for x:
x = 2y + 11.

Step 2: Substitute the expression found in step 1 (x = 2y + 11) into the other equation. The other equation is −7x − 2y = −13. Substitute x with 2y + 11:
−7(2y + 11) − 2y = −13.

Step 3: Distribute the −7 to the terms inside the parentheses:
−14y − 77 − 2y = −13.

Step 4: Combine like terms on the left-hand side of the equation:
−16y − 77 = −13.

Step 5: Isolate the variable y by adding 77 to both sides of the equation:
−16y = 64.

Step 6: Divide both sides of the equation by −16 to solve for y:
y = −4.

Step 7: Substitute the value of y back into one of the original equations to solve for x. We'll use x − 2y = 11. Substitute y with −4:
x − 2(-4) = 11.

Step 8: Simplify the equation:
x + 8 = 11.

Step 9: Isolate the variable x by subtracting 8 from both sides of the equation:
x = 3.

Therefore, the solution to the system of equations −7x − 2y = −13 and x − 2y = 11 is x = 3 and y = −4.

Answer 3

[AI-generated answer — Explain Bot]

To solve a system of equations by substitution, follow these steps:

Step 1: Choose one of the equations and solve it for one variable in terms of the other variable.

In this case, let's choose the second equation, x - 2y = 11, and solve it for x:
x = 2y + 11

Step 2: Substitute the expression found in Step 1 into the other equation.

Substitute the expression x = 2y + 11 into the first equation, -7x - 2y = -13:
-7(2y + 11) - 2y = -13

Step 3: Simplify and solve the resulting equation for the remaining variable.

-14y - 77 - 2y = -13
-16y - 77 = -13
-16y = 64
y = 4

Step 4: Substitute the value of the solved variable back into one of the original equations to find the value of the other variable.

Substitute y = 4 into x - 2y = 11:
x - 2(4) = 11
x - 8 = 11
x = 19

Step 5: Check the solution by substituting the values of x and y into both equations of the system.

Substitute x = 19 and y = 4 into the first equation, -7x - 2y = -13:
-7(19) - 2(4) = -13
-133 - 8 = -13
-141 = -13 (not true)

Substitute x = 19 and y = 4 into the second equation, x - 2y = 11:
19 - 2(4) = 11
19 - 8 = 11
11 = 11 (true)

Since the solution satisfies only one of the equations, this system of equations is inconsistent and has no solution.