Questions: Solve the system of equations by the substitution method.
4x + 3y = 27
y = -(4/3)x + 9
(x, y) = ( )
Transcript text: Solve the system of equations by the substitution method.
\[
\begin{array}{l}
\left\{\begin{array}{r}
4 x+3 y=27 \\
y=-\frac{4}{3} x+9
\end{array}\right. \\
(x, y)=(\square)
\end{array}
\]
Solution
Solution Steps
To solve the system of equations using the substitution method, follow these steps:
Substitute the expression for y from the second equation into the first equation.
Solve the resulting equation for x.
Substitute the value of x back into the second equation to find y.
Step 1: Substitute the Expression for y
Given the system of equations:
{4x+3y=27y=−34x+9
We will use the substitution method. First, substitute the expression for y from the second equation into the first equation.
Step 2: Substitute and Simplify
Substitute y=−34x+9 into 4x+3y=27:
4x+3(−34x+9)=27
Simplify inside the parentheses:
4x+3(−34x)+3⋅9=27
4x−4x+27=27
Step 3: Solve for x
Combine like terms:
0x+27=27
This simplifies to:
27=27
This is a true statement, which means the system of equations is dependent, and there are infinitely many solutions. The equations represent the same line.
Final Answer
Since the system is dependent, the solution is the set of all points (x,y) that satisfy the equation y=−34x+9.