To find the solution to the given system of equations, we can use a method such as substitution or elimination. Here, we'll use the substitution method for the detailed step-by-step solution.
We start with the given system of equations:
[tex]\[
\begin{array}{l}
2x = -y + 6 \\
-4x + 3y = 8
\end{array}
\][/tex]
### Step 1: Solve the first equation for [tex]\( y \)[/tex]:
First, solve the first equation for [tex]\( y \)[/tex]:
[tex]\[
2x = -y + 6
\][/tex]
Add [tex]\( y \)[/tex] to both sides:
[tex]\[
2x + y = 6
\][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 6 - 2x
\][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] in the second equation:
Now substitute [tex]\( y = 6 - 2x \)[/tex] into the second equation:
[tex]\[
-4x + 3y = 8
\][/tex]
Replace [tex]\( y \)[/tex] with [tex]\( 6 - 2x \)[/tex]:
[tex]\[
-4x + 3(6 - 2x) = 8
\][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]:
Expand the equation:
[tex]\[
-4x + 18 - 6x = 8
\][/tex]
Combine like terms:
[tex]\[
-10x + 18 = 8
\][/tex]
Subtract 18 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
-10x = 8 - 18
\][/tex]
[tex]\[
-10x = -10
\][/tex]
Divide both sides by [tex]\(-10\)[/tex]:
[tex]\[
x = 1
\][/tex]
### Step 4: Find [tex]\( y \)[/tex] using the value of [tex]\( x \)[/tex]:
Now substitute [tex]\( x = 1 \)[/tex] back into the equation [tex]\( y = 6 - 2x \)[/tex]:
[tex]\[
y = 6 - 2(1)
\][/tex]
[tex]\[
y = 6 - 2
\][/tex]
[tex]\[
y = 4
\][/tex]
### Conclusion:
The solution to the system of equations is [tex]\( x = 1 \)[/tex] and [tex]\( y = 4 \)[/tex]. Therefore, the point [tex]\((1, 4)\)[/tex] is the solution.
From the given options, we can see that the correct answer is:
[tex]\[
(1, 4)
\][/tex]
