Question
Solve the system by substitution. \[ \begin{aligned} y & =3 x+28 \\ y & =-4 x\end{aligned} \]
Ask by Alexander Ortega. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( (x,y) = (-4,16) \)
Solution
Solve the system of equations \( y=3x+28;y=-4x \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}y=3x+28\\y=-4x\end{array}\right.\)
- step1: Substitute the value of \(y:\)
\(3x+28=-4x\)
- step2: Move the variable to the left side:
\(3x+28+4x=0\)
- step3: Add the terms:
\(7x+28=0\)
- step4: Move the constant to the right side:
\(7x=0-28\)
- step5: Remove 0:
\(7x=-28\)
- step6: Divide both sides:
\(\frac{7x}{7}=\frac{-28}{7}\)
- step7: Divide the numbers:
\(x=-4\)
- step8: Substitute the value of \(x:\)
\(y=3\left(-4\right)+28\)
- step9: Simplify:
\(y=16\)
- step10: Calculate:
\(\left\{ \begin{array}{l}x=-4\\y=16\end{array}\right.\)
- step11: Check the solution:
\(\left\{ \begin{array}{l}x=-4\\y=16\end{array}\right.\)
- step12: Rewrite:
\(\left(x,y\right) = \left(-4,16\right)\)
The solution to the system of equations by substitution is \( (x,y) = (-4,16) \).
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Simplify this solution Extra Insights
To solve the system by substitution, we start with the two equations provided: 1. \( y = 3x + 28 \) 2. \( y = -4x \) Since both equations equal \( y \), we can set them equal to each other: \[ 3x + 28 = -4x \] Now, we'll solve for \( x \) by adding \( 4x \) to both sides: \[ 3x + 4x + 28 = 0 \] \[ 7x + 28 = 0 \] Next, subtract 28 from both sides: \[ 7x = -28 \] Now, divide both sides by 7 to find \( x \): \[ x = -4 \] Now that we have the value for \( x \), we can substitute it back into either original equation to find \( y \). We'll use the second equation for simplicity: \[ y = -4(-4) = 16 \] Thus, the solution to the system of equations is: \[ (x, y) = (-4, 16) \]
