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Review the graphs of a system of two linear equations in two variables: 4x+3y=15 and y=x+5. Find the solution to both equations...Asked by Kadyn
Review the graphs of a system of two linear equations in two variables: x−5y=7 and 3x−2y=−4 . Find the solution to both equations
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Answered by
GPT-4o mini
To find the solution to the system of equations given by:
1. \( x - 5y = 7 \)
2. \( 3x - 2y = -4 \)
we can use either the substitution method or the elimination method. Here, I'll use the substitution method.
### Step 1: Solve the first equation for \( x \)
From the first equation:
\[
x = 5y + 7
\]
### Step 2: Substitute for \( x \) in the second equation
Now, substitute \( x \) from the first equation into the second equation:
\[
3(5y + 7) - 2y = -4
\]
### Step 3: Distribute and simplify
Distributing \( 3 \):
\[
15y + 21 - 2y = -4
\]
Combine like terms:
\[
(15y - 2y) + 21 = -4
\]
\[
13y + 21 = -4
\]
### Step 4: Isolate \( y \)
Subtract \( 21 \) from both sides:
\[
13y = -4 - 21
\]
\[
13y = -25
\]
Now, divide both sides by \( 13 \):
\[
y = -\frac{25}{13}
\]
### Step 5: Substitute back to find \( x \)
Substitute \( y \) back into the equation \( x = 5y + 7 \):
\[
x = 5\left(-\frac{25}{13}\right) + 7
\]
Calculating \( 5 \cdot -\frac{25}{13} \):
\[
x = -\frac{125}{13} + 7
\]
\[
x = -\frac{125}{13} + \frac{91}{13}
\]
\[
x = -\frac{125 - 91}{13}
\]
\[
x = -\frac{34}{13}
\]
### Final Solution
The solution to the system of equations is:
\[
\left( -\frac{34}{13}, -\frac{25}{13} \right)
\]
This means that the point \( \left( -\frac{34}{13}, -\frac{25}{13} \right) \) is the point where the two lines intersect, which is the solution to the system.
1. \( x - 5y = 7 \)
2. \( 3x - 2y = -4 \)
we can use either the substitution method or the elimination method. Here, I'll use the substitution method.
### Step 1: Solve the first equation for \( x \)
From the first equation:
\[
x = 5y + 7
\]
### Step 2: Substitute for \( x \) in the second equation
Now, substitute \( x \) from the first equation into the second equation:
\[
3(5y + 7) - 2y = -4
\]
### Step 3: Distribute and simplify
Distributing \( 3 \):
\[
15y + 21 - 2y = -4
\]
Combine like terms:
\[
(15y - 2y) + 21 = -4
\]
\[
13y + 21 = -4
\]
### Step 4: Isolate \( y \)
Subtract \( 21 \) from both sides:
\[
13y = -4 - 21
\]
\[
13y = -25
\]
Now, divide both sides by \( 13 \):
\[
y = -\frac{25}{13}
\]
### Step 5: Substitute back to find \( x \)
Substitute \( y \) back into the equation \( x = 5y + 7 \):
\[
x = 5\left(-\frac{25}{13}\right) + 7
\]
Calculating \( 5 \cdot -\frac{25}{13} \):
\[
x = -\frac{125}{13} + 7
\]
\[
x = -\frac{125}{13} + \frac{91}{13}
\]
\[
x = -\frac{125 - 91}{13}
\]
\[
x = -\frac{34}{13}
\]
### Final Solution
The solution to the system of equations is:
\[
\left( -\frac{34}{13}, -\frac{25}{13} \right)
\]
This means that the point \( \left( -\frac{34}{13}, -\frac{25}{13} \right) \) is the point where the two lines intersect, which is the solution to the system.
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