Certainly! Let's solve this system of equations step-by-step using substitution and identify the solution.
Given system of equations:
[tex]\[
\begin{cases}
y = -3x + 3 \\
7x + 2y = 1
\end{cases}
\][/tex]
### Step 1: Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]
The first equation already expresses [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = -3x + 3 \][/tex]
### Step 2: Substitute this expression into the second equation
Substitute [tex]\( y = -3x + 3 \)[/tex] into the second equation [tex]\( 7x + 2y = 1 \)[/tex]:
[tex]\[ 7x + 2(-3x + 3) = 1 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
Simplify the equation:
[tex]\[ 7x + 2(-3x) + 2(3) = 1 \][/tex]
[tex]\[ 7x - 6x + 6 = 1 \][/tex]
[tex]\[ x + 6 = 1 \][/tex]
Subtract 6 from both sides:
[tex]\[ x = 1 - 6 \][/tex]
[tex]\[ x = -5 \][/tex]
### Step 4: Substitute [tex]\( x = -5 \)[/tex] back into the first equation
To find [tex]\( y \)[/tex], substitute [tex]\( x = -5 \)[/tex] back into the first equation [tex]\( y = -3x + 3 \)[/tex]:
[tex]\[ y = -3(-5) + 3 \][/tex]
[tex]\[ y = 15 + 3 \][/tex]
[tex]\[ y = 18 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y) = (-5, 18) \][/tex]
### Step 5: Verify using the points given
The given points are:
[tex]\[
(18, -5), (2, 3), (4, -9), (-5, 18), (-9, 4), (3, 2)
\][/tex]
We need to check which of these points are the solution to the system of equations:
[tex]\[
\begin{cases}
y = -3x + 3 \\
7x + 2y = 1
\end{cases}
\][/tex]
The solution we found through substitution is [tex]\( (-5, 18) \)[/tex], which matches one of the given points. Therefore, the point that satisfies both equations is:
[tex]\[ (-5, 18) \][/tex]
Thus, the solution to the system of equations, verified against the given points, is:
[tex]\[ \boxed{(-5, 18)} \][/tex]
