Answer :
Certainly! Let's solve the given system of linear equations step-by-step:
[tex]\[ \begin{cases} 2x + 3y = 5 \\ 2x - y = -7 \end{cases} \][/tex]
### Step 1: Express [tex]\( y \)[/tex] from the second equation.
First, we solve the second equation for [tex]\( y \)[/tex]:
[tex]\[ 2x - y = -7 \][/tex]
Rearrange to solve for [tex]\( y \)[/tex]:
[tex]\[ -y = -7 - 2x \][/tex]
Multiply by [tex]\(-1\)[/tex] to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 7 + 2x \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into the first equation.
Now we substitute [tex]\( y = 7 + 2x \)[/tex] into the first equation:
[tex]\[ 2x + 3(7 + 2x) = 5 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex].
Distribute the 3 in the equation:
[tex]\[ 2x + 21 + 6x = 5 \][/tex]
Combine like terms:
[tex]\[ 8x + 21 = 5 \][/tex]
Subtract 21 from both sides:
[tex]\[ 8x = 5 - 21 \][/tex]
[tex]\[ 8x = -16 \][/tex]
Divide by 8:
[tex]\[ x = -2 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex].
Substitute [tex]\( x = -2 \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[ y = 7 + 2(-2) \][/tex]
Simplify:
[tex]\[ y = 7 - 4 \][/tex]
[tex]\[ y = 3 \][/tex]
### Solution
The solution to the system of equations is:
[tex]\[ (x, y) = (-2, 3) \][/tex]
[tex]\[ \begin{cases} 2x + 3y = 5 \\ 2x - y = -7 \end{cases} \][/tex]
### Step 1: Express [tex]\( y \)[/tex] from the second equation.
First, we solve the second equation for [tex]\( y \)[/tex]:
[tex]\[ 2x - y = -7 \][/tex]
Rearrange to solve for [tex]\( y \)[/tex]:
[tex]\[ -y = -7 - 2x \][/tex]
Multiply by [tex]\(-1\)[/tex] to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 7 + 2x \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into the first equation.
Now we substitute [tex]\( y = 7 + 2x \)[/tex] into the first equation:
[tex]\[ 2x + 3(7 + 2x) = 5 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex].
Distribute the 3 in the equation:
[tex]\[ 2x + 21 + 6x = 5 \][/tex]
Combine like terms:
[tex]\[ 8x + 21 = 5 \][/tex]
Subtract 21 from both sides:
[tex]\[ 8x = 5 - 21 \][/tex]
[tex]\[ 8x = -16 \][/tex]
Divide by 8:
[tex]\[ x = -2 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex].
Substitute [tex]\( x = -2 \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[ y = 7 + 2(-2) \][/tex]
Simplify:
[tex]\[ y = 7 - 4 \][/tex]
[tex]\[ y = 3 \][/tex]
### Solution
The solution to the system of equations is:
[tex]\[ (x, y) = (-2, 3) \][/tex]