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Completion requirements
Solve the given system of equations by substitution. Verify the solution.
\(\left\{ \begin{array}{l}
y = 2x^2 - 14x + 55 \\
y = 4x^2 + 15x - 30 \end{array} \right.\)
\(\left\{ \begin{array}{l}
y = 2x^2 - 14x + 55 \\
y = 4x^2 + 15x - 30 \end{array} \right.\)
The \(y\) is isolated in the first equation, so substitute its equivalent for \(y\) in the second equation and solve.
\(\begin{align}
y &= 4x^2 + 15x - 30 \\
2x^2 - 14x + 55 &= 4x^2 + 15x - 30 \\
0 &= 2x^2 + 29x - 85 \\
0 &= 2x^2 + 34x - 5x - 85 \\
0 &= 2x\left( {x + 17} \right) - 5\left( {x + 17} \right) \\
0 &= \left( {x + 17} \right)\left( {2x - 5} \right) \end{align}\)
Substitute each \(x\)-value into any equation that contains both \(x\) and \(y\) to determine the \(y\)-value.
The solutions to the system are \(\left( { - 17,871} \right)\) and \(\left( {\frac{5}{2},\frac{{65}}{2}} \right)\).
Verify for \((-17, 871)\).
Verify for \(\left( {\frac{5}{2},\frac{{65}}{2}} \right)\).
\(\begin{align}
y &= 4x^2 + 15x - 30 \\
2x^2 - 14x + 55 &= 4x^2 + 15x - 30 \\
0 &= 2x^2 + 29x - 85 \\
0 &= 2x^2 + 34x - 5x - 85 \\
0 &= 2x\left( {x + 17} \right) - 5\left( {x + 17} \right) \\
0 &= \left( {x + 17} \right)\left( {2x - 5} \right) \end{align}\)
\(\begin{align}
x + 17 &= 0 \\
x &= - 17 \end{align}\)
x + 17 &= 0 \\
x &= - 17 \end{align}\)
\(\begin{align}
2x - 5 &= 0 \\
2x &= 5 \\
x &= \frac{5}{2}
\end{align}\)
2x - 5 &= 0 \\
2x &= 5 \\
x &= \frac{5}{2}
\end{align}\)
Substitute each \(x\)-value into any equation that contains both \(x\) and \(y\) to determine the \(y\)-value.
\[\begin{align}
y &= 2x^2 - 14x + 55 \\
y &= 2\left( { - 17} \right)^2 - 14\left( { - 17} \right) + 55 \\
y &= 871 \end{align}\]
y &= 2x^2 - 14x + 55 \\
y &= 2\left( { - 17} \right)^2 - 14\left( { - 17} \right) + 55 \\
y &= 871 \end{align}\]
\[\begin{align}
y &= 2x^2 - 14x + 55 \\
y &= 2\left( {\frac{5}{2}} \right)^2 - 14\left( {\frac{5}{2}} \right) + 55 \\
y &= \frac{{65}}{2} \end{align}\]
y &= 2x^2 - 14x + 55 \\
y &= 2\left( {\frac{5}{2}} \right)^2 - 14\left( {\frac{5}{2}} \right) + 55 \\
y &= \frac{{65}}{2} \end{align}\]
The solutions to the system are \(\left( { - 17,871} \right)\) and \(\left( {\frac{5}{2},\frac{{65}}{2}} \right)\).
Verify for \((-17, 871)\).
\(y = 2x^2 - 14x + 55\)
| Left Side | Right Side |
|---|---|
|
\(\begin{array}{r} y \\ 871 \end{array}\) |
\(\begin{array} 2x^2 - 14x + 55 \\ 2\left( { - 17} \right)^2 - 14\left( { - 17} \right) + 55 \\ 871 \end{array}\) |
| \(\hspace{25pt}\)LS = RS | |
\(y = 4x^2 + 15x - 30\)
| Left Side | Right Side |
|---|---|
|
\(\begin{array}{r} y \\ 871 \end{array}\) |
\(\begin{array} 4x^2 + 15x - 30 \\ 4\left( { - 17} \right)^2 + 15\left( { - 17} \right) - 30 \\ 871 \end{array}\) |
| \(\hspace{25pt}\)LS = RS | |
Verify for \(\left( {\frac{5}{2},\frac{{65}}{2}} \right)\).
\(y = 2x^2 - 14x + 55\)
| Left Side | Right Side |
|---|---|
| \[\begin{array}{r} y \\ \frac{{65}}{2} \end{array}\] |
\[\begin{array}{l} 2x^2 - 14x + 55 \\ 2\left( {\frac{5}{2}} \right)^2 - 14\left( {\frac{5}{2}} \right) + 55 \\ \frac{{65}}{2} \end{array}\] |
| \(\hspace{25pt}\)LS = RS | |
\(y = 4x^2 + 15x - 30\)
| Left Side | Right Side |
|---|---|
|
\[\begin{array}{r} y \\ \frac{{65}}{2} \end{array}\] |
\[\begin{array}{l} 4x^2 + 15x - 30 \\ 4\left( {\frac{5}{2}} \right)^2 + 15\left( {\frac{5}{2}} \right) - 30 \\ \frac{{65}}{2} \end{array}\] |
| \(\hspace{25pt}\)LS = RS | |