Mejia Chan
05/19/2024 · Senior High School
Solve the following system of equations graphically on the set of axes below. \[ \begin{array}{c}y=\frac{3}{4} x+6 \\ y=-2 x-5\end{array} \]
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Step-by-step Solution
Solve the system of equations \( y=\frac{3}{4}x+6;y=-2x-5 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}y=\frac{3}{4}x+6\\y=-2x-5\end{array}\right.\)
- step1: Substitute the value of \(y:\)
\(\frac{3}{4}x+6=-2x-5\)
- step2: Move the variable to the left side:
\(\frac{3}{4}x+6+2x=-5\)
- step3: Add the terms:
\(\frac{11}{4}x+6=-5\)
- step4: Move the constant to the right side:
\(\frac{11}{4}x=-5-6\)
- step5: Subtract the numbers:
\(\frac{11}{4}x=-11\)
- step6: Multiply by the reciprocal:
\(\frac{11}{4}x\times \frac{4}{11}=-11\times \frac{4}{11}\)
- step7: Multiply:
\(x=-4\)
- step8: Substitute the value of \(x:\)
\(y=\frac{3}{4}\left(-4\right)+6\)
- step9: Simplify:
\(y=3\)
- step10: Calculate:
\(\left\{ \begin{array}{l}x=-4\\y=3\end{array}\right.\)
- step11: Check the solution:
\(\left\{ \begin{array}{l}x=-4\\y=3\end{array}\right.\)
- step12: Rewrite:
\(\left(x,y\right) = \left(-4,3\right)\)
The solution to the system of equations is \( (x, y) = (-4, 3) \). This point represents the intersection of the two lines on the graph.
Quick Answer
The solution is \( (x, y) = (-4, 3) \).
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