Question
Solve the system of equations 2, x, minus, 2, y, equals, 02x−2y=0 and minus, 5, x, plus, 7, y, equals, minus, 12−5x+7y=−12 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.
2
222222222222222 left parenthesis, 2, x(2x minus, 2, y−2y equals= 0, right parenthesis0)
-5
-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5 left parenthesis, minus, 5, x(−5x plus, 7, y+7y equals= minus, 12, right parenthesis−12)
4, x4x minus, 4, y−4y equals= 00
25, x25x minus, 35, y−35y equals= 6060
xx plus+
yy 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.
2
222222222222222 left parenthesis, 2, x(2x minus, 2, y−2y equals= 0, right parenthesis0)
-5
-5-5-5-5-5-5-5-5-5-5-5-5-5-5-5 left parenthesis, minus, 5, x(−5x plus, 7, y+7y equals= minus, 12, right parenthesis−12)
4, x4x minus, 4, y−4y equals= 00
25, x25x minus, 35, y−35y equals= 6060
xx plus+
yy equals=
Answers
Answered by
GPT-4o mini
To solve the system of equations by combining them, we start with the given equations:
1. \( 2x - 2y = 0 \)
2. \( -5x + 7y = -12 \)
### Step 1: Multiply the equations
We'll multiply the first equation by \( 2.5 \) so that the coefficient of \( x \) in the first equation matches the coefficient of \( x \) in the second equation.
**Multiplying the first equation by \( 2.5 \)**:
\[
2.5 \times (2x - 2y) = 2.5 \times 0
\]
Calculating this gives:
\[
5x - 5y = 0
\]
### New system of equations:
1. \( 5x - 5y = 0 \) (new version of first equation)
2. \( -5x + 7y = -12 \)
### Step 2: Add the equations
Now we can add the two equations together:
\[
(5x - 5y) + (-5x + 7y) = 0 + (-12)
\]
This simplifies to:
\[
0 + 2y = -12
\]
### Step 3: Solve for \( y \)
\[
2y = -12
\]
Dividing by \( 2 \):
\[
y = -6
\]
### Step 4: Substitute back to find \( x \)
Now substitute \( y = -6 \) back into one of the original equations to find \( x \). We will use the first equation:
\[
2x - 2(-6) = 0
\]
This becomes:
\[
2x + 12 = 0
\]
Subtracting \( 12 \) from both sides:
\[
2x = -12
\]
Dividing by \( 2 \):
\[
x = -6
\]
### Final Solution
The solution to the system of equations is:
\[
x = -6, \quad y = -6
\]
So the final answer is:
\[
(x, y) = (-6, -6)
\]
1. \( 2x - 2y = 0 \)
2. \( -5x + 7y = -12 \)
### Step 1: Multiply the equations
We'll multiply the first equation by \( 2.5 \) so that the coefficient of \( x \) in the first equation matches the coefficient of \( x \) in the second equation.
**Multiplying the first equation by \( 2.5 \)**:
\[
2.5 \times (2x - 2y) = 2.5 \times 0
\]
Calculating this gives:
\[
5x - 5y = 0
\]
### New system of equations:
1. \( 5x - 5y = 0 \) (new version of first equation)
2. \( -5x + 7y = -12 \)
### Step 2: Add the equations
Now we can add the two equations together:
\[
(5x - 5y) + (-5x + 7y) = 0 + (-12)
\]
This simplifies to:
\[
0 + 2y = -12
\]
### Step 3: Solve for \( y \)
\[
2y = -12
\]
Dividing by \( 2 \):
\[
y = -6
\]
### Step 4: Substitute back to find \( x \)
Now substitute \( y = -6 \) back into one of the original equations to find \( x \). We will use the first equation:
\[
2x - 2(-6) = 0
\]
This becomes:
\[
2x + 12 = 0
\]
Subtracting \( 12 \) from both sides:
\[
2x = -12
\]
Dividing by \( 2 \):
\[
x = -6
\]
### Final Solution
The solution to the system of equations is:
\[
x = -6, \quad y = -6
\]
So the final answer is:
\[
(x, y) = (-6, -6)
\]