Put the steps in order that are used to solve the following systems of equations by substitution. −7x−2y%3D−13 x−2y%3D11
1. Solve one of the equations for one variable in terms of the other variable.
2. Substitute the expression found in step 1 into the other equation.
3. Solve the resulting equation for the remaining variable.
4. Substitute the value found in step 3 back into one of the original equations to solve for the other variable.
5. Check the solution by substituting the values found in step 4 into both equations to ensure they satisfy the equations.
To solve the system of equations −7x−2y=−13 and x−2y=11 by substitution, here are the steps in order:
Step 1: Choose one of the equations to solve for one variable in terms of the other variable.
In this case, we will solve the second equation x−2y=11 for x in terms of y:
x = 11 + 2y
Step 2: Substitute the expression from step 1 into the remaining equation.
Substitute x = 11 + 2y into the first equation −7x−2y=−13:
−7(11 + 2y) − 2y = −13
Step 3: Simplify and solve the resulting equation.
−77 - 14y - 2y = −13
-77 - 16y = -13
-16y = -13 + 77
-16y = 64
y = 64 / -16
y = -4
Step 4: Substitute the value of y back into one of the original equations to solve for the other variable.
Substitute y = -4 into the second equation x−2y=11:
x−2(-4) = 11
x + 8 = 11
x = 11 - 8
x = 3
Step 5: Check the solution by substituting the values of x and y back into both original equations.
Plug x = 3 and y = -4 into the first equation −7x−2y=−13:
−7(3)−2(-4) = −13
-21 + 8 = -13
-13 = -13 (True)
Plug x = 3 and y = -4 into the second equation x−2y=11:
3 − 2(-4) = 11
3 + 8 = 11
11 = 11 (True)
Therefore, the solution to the given system of equations is x = 3 and y = -4.
To solve the system of equations by substitution between −7x−2y=−13 and x−2y=11, follow these steps:
1. Choose one of the equations to solve for one variable in terms of the other variable. In this case, we can solve the second equation, x−2y=11, for x:
- Add 2y to both sides of the equation: x = 2y + 11.
2. Substitute the expression found in step 1 into the other equation. Replace the x value in the first equation, −7x−2y=−13, with 2y + 11:
- Replace x with 2y + 11 in the first equation: −7(2y + 11) − 2y = −13.
3. Simplify and solve the resulting equation. Distribute the -7 to the terms inside the parentheses:
- Expanding the equation: -14y - 77 - 2y = -13.
- Combine like terms: -16y - 77 = -13.
- Add 77 to both sides to isolate the variable term: -16y = 64.
- Divide both sides by -16 to solve for y: y = -4.
4. Substitute the value of y back into either of the original equations. We will use the second equation, x − 2y = 11:
- Replace y with -4 in x - 2y = 11: x - 2(-4) = 11.
- Simplify: x + 8 = 11.
- Subtract 8 from both sides to solve for x: x = 3.
5. Check the solution by substituting the found values of x and y into the second equation: x - 2y = 11:
- Substitute x = 3 and y = -4 into x - 2y = 11: 3 - 2(-4) = 11.
- Simplify: 3 + 8 = 11.
- The equation is true, so the solution (x, y) = (3, -4) is valid.
Therefore, the solution to the system of equations −7x−2y=−13 and x−2y=11 is (x, y) = (3, -4).