Question
Solve the system of equations minus, 2, x, minus, 7, y, equals, minus, 14−2x−7y=−14 and minus, x, minus, 5, y, equals, minus, 13−x−5y=−13 by combining the equations.
Below, two sets of the two equations are shown. The first set has input boxes in front of each equation to put a number to multiply it by. When you enter a number, the corresponding equation below changes to be the result of the multiplication. Under the second set, a resulting equation is shown with input boxes for the coefficients and constant. Then, there is a "try" button. When you press the try button, if your answer is correct, it will be placed on the page and a new question will appear with a new input box. This process will continue until the final answer area appears with the submit button.
left parenthesis, minus, 2, x(−2x minus, 7, y−7y equals= minus, 14, right parenthesis−14)
left parenthesis, minus, x(−x minus, 5, y−5y equals= minus, 13, right parenthesis−13)
minus, 2, x−2x minus, 7, y−7y equals= minus, 14−14
minus, x−x minus, 5, y−5y equals= minus, 13−13
0
0xx plus+
0
0yy equals=
Below, two sets of the two equations are shown. The first set has input boxes in front of each equation to put a number to multiply it by. When you enter a number, the corresponding equation below changes to be the result of the multiplication. Under the second set, a resulting equation is shown with input boxes for the coefficients and constant. Then, there is a "try" button. When you press the try button, if your answer is correct, it will be placed on the page and a new question will appear with a new input box. This process will continue until the final answer area appears with the submit button.
left parenthesis, minus, 2, x(−2x minus, 7, y−7y equals= minus, 14, right parenthesis−14)
left parenthesis, minus, x(−x minus, 5, y−5y equals= minus, 13, right parenthesis−13)
minus, 2, x−2x minus, 7, y−7y equals= minus, 14−14
minus, x−x minus, 5, y−5y equals= minus, 13−13
0
0xx plus+
0
0yy equals=
Answers
GPT-4o mini
To solve the system of equations by combining them, we will first rewrite the two equations:
1) \( -2x - 7y = -14 \) (Equation 1)
2) \( -x - 5y = -13 \) (Equation 2)
Next, we can manipulate these equations to eliminate one of the variables. Let's first multiply Equation 2 by 2 to align the coefficients of \( x \):
\[
2(-x - 5y) = 2(-13)
\]
This gives us:
\[
-2x - 10y = -26 \quad (Equation 3)
\]
Now we have the system:
1) \( -2x - 7y = -14 \)
3) \( -2x - 10y = -26 \)
Now we can eliminate \( -2x \) from both equations. We do this by subtracting Equation 1 from Equation 3:
\[
(-2x - 10y) - (-2x - 7y) = -26 - (-14)
\]
This simplifies to:
\[
-10y + 7y = -26 + 14
\]
So, we get:
\[
-3y = -12
\]
Now, divide both sides by -3:
\[
y = 4
\]
Now that we have \( y \), we can substitute it back into one of the original equations to find \( x \). Let's substitute \( y = 4 \) into Equation 1:
\[
-2x - 7(4) = -14
\]
This simplifies to:
\[
-2x - 28 = -14
\]
Adding 28 to both sides gives:
\[
-2x = 14
\]
Dividing by -2, we find:
\[
x = -7
\]
So, the solution to the system of equations is:
\[
x = -7, \quad y = 4
\]
The final answer is:
\[
(x, y) = (-7, 4)
\]
1) \( -2x - 7y = -14 \) (Equation 1)
2) \( -x - 5y = -13 \) (Equation 2)
Next, we can manipulate these equations to eliminate one of the variables. Let's first multiply Equation 2 by 2 to align the coefficients of \( x \):
\[
2(-x - 5y) = 2(-13)
\]
This gives us:
\[
-2x - 10y = -26 \quad (Equation 3)
\]
Now we have the system:
1) \( -2x - 7y = -14 \)
3) \( -2x - 10y = -26 \)
Now we can eliminate \( -2x \) from both equations. We do this by subtracting Equation 1 from Equation 3:
\[
(-2x - 10y) - (-2x - 7y) = -26 - (-14)
\]
This simplifies to:
\[
-10y + 7y = -26 + 14
\]
So, we get:
\[
-3y = -12
\]
Now, divide both sides by -3:
\[
y = 4
\]
Now that we have \( y \), we can substitute it back into one of the original equations to find \( x \). Let's substitute \( y = 4 \) into Equation 1:
\[
-2x - 7(4) = -14
\]
This simplifies to:
\[
-2x - 28 = -14
\]
Adding 28 to both sides gives:
\[
-2x = 14
\]
Dividing by -2, we find:
\[
x = -7
\]
So, the solution to the system of equations is:
\[
x = -7, \quad y = 4
\]
The final answer is:
\[
(x, y) = (-7, 4)
\]
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