−7x−2y=−13 x−2y=11
Show all your steps for solving this equation.
To solve this system of equations, we can use the method of substitution or elimination. Let's solve it using the substitution method:
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation:
x - 2y = 11
We can rearrange this equation to solve for x:
x = 2y + 11
Step 2: Substitute the expression for the solved variable in the other equation.
Now we substitute the expression for x in the first equation:
-7(2y + 11) - 2y = -13
Step 3: Solve the resulting equation for the remaining variable.
We simplify the equation:
-14y - 77 - 2y = -13
-16y - 77 = -13
-16y = 77 - 13
-16y = 64
y = 64 / -16
y = -4
Step 4: Substitute the value of y back into one of the original equations to solve for x.
Using the second equation:
x - 2y = 11
x - 2(-4) = 11
x + 8 = 11
x = 11 - 8
x = 3
Step 5: Check the solution.
Plug the values of x and y into the first equation:
-7x - 2y = -13
-7(3) - 2(-4) = -13
-21 + 8 = -13
-13 = -13
The solution to the system of equations is x = 3, y = -4.
To solve this system of equations, we can use the method of elimination. The goal is to eliminate one variable so we can solve for the other.
Step 1: Start by multiplying both sides of the second equation by -7 to make the x-coefficients in both equations opposites of each other:
-7(x - 2y) = -7(11)
This simplifies to:
-7x + 14y = -77
Step 2: Now we can add the two equations together to eliminate the y variable:
(-7x - 2y) + (-7x + 14y) = -13 + (-77)
Simplifying this gives us:
-14x + 12y = -90
Step 3: We now have a new equation with only one variable, x. Solve for x by isolating it:
-14x = -90 - 12y
Divide both sides by -14:
x = (-90 - 12y) / -14
This can be simplified further if needed.
Step 4: Substitute the value of x in terms of y back into one of the original equations. Let's use the second equation:
x - 2y = 11
(-90 - 12y) / -14 - 2y = 11
Now we can solve this equation for y:
Step 5: Simplify and solve for y:
(-90 - 12y - 28y) / -14 = 11
-90 - 40y = 11 * -14
-90 - 40y = -154
Add 90 to both sides:
-40y = -64
Divide both sides by -40:
y = -64 / -40
Simplify the fraction if needed.
Step 6: Substitute the value of y back into either of the original equations to solve for x. Using the first equation:
-7x - 2(-64/-40) = -13
-7x + 128/40 = -13
Simplify and solve for x:
Step 7: Add 13 to both sides:
-7x + 128/40 + 13 = 0
-7x + 128/40 + 520/40 = 0
Combine the fractions on the left side:
-7x + 648/40 = 0
Step 8: Convert the fraction to a decimal:
-7x + 16.2 = 0
Step 9: Subtract 16.2 from both sides to isolate x:
-7x = -16.2
Divide both sides by -7:
x = -16.2 / -7
Simplify the fraction if needed.
So the solution to the system of equations is x = -2.314... and y = 1.6.
To solve this system of equations, we can use the method of elimination. Here are the steps:
Step 1: Multiply the second equation by -7 to match the coefficients of 'x' with the coefficients in the first equation.
-7(x - 2y) = -7(11)
-7x + 14y = -77
Step 2: Now we can add the two equations together, eliminating the variable 'x'.
(-7x - 2y) + (-7x + 14y) = (-13) + (-77)
-14x + 12y = -90
Step 3: Simplify the equation.
-14x + 12y = -90
Step 4: Divide the equation by -2 to solve for 'y'.
(-14x + 12y)/(-2) = -90/(-2)
7x - 6y = 45
Step 5: Now we have two equations:
-14x + 12y = -90
7x - 6y = 45
Step 6: Add the two equations together to eliminate the variable 'y'.
(-14x + 12y) + (7x - 6y) = -90 + 45
-7x + 6y = -45
Step 7: Multiply the equation by 2 to solve for 'x'.
2(-7x + 6y) = 2(-45)
-14x + 12y = -90
Step 8: Since we obtained the same equation as in Step 3, this implies that the two equations in the system are dependent. This means that the system has infinitely many solutions.
So, the solution to the system of equations is all the values of 'x' and 'y' that satisfy the equation -14x + 12y = -90.