Question
Solving Systems by Substitution Solve each system by substitution. 1) \[ \begin{array}{l} x=-2 y \\ x-y=9 \end{array} \] 3) \[ \begin{array}{l} -x-3 y=-18 \\ y=-4 x+6 \end{array} \] 5) \[ \begin{array}{l} y=-6 \\ 3 x-6 y=30 \end{array} \] 2) \[ \begin{array}{l} y=x-4 \\ -4 x-6 y=-16 \end{array} \] 4) \[ \begin{aligned} -2 x+2 y & =6 \\ 4 x+2 y & =-5 \end{aligned} \] G) \[ \begin{aligned} -2 x-y & =-5 \\ x-8 y & =-23 \end{aligned} \]
Ask by Fernandez George. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
1. \( (6, -3) \)
2. \( (4, 0) \)
3. \( (0, 6) \)
4. \( \left(-\frac{11}{6}, \frac{7}{6}\right) \)
5. \( (-2, -6) \)
6. \( (1, 3) \)
Solution
**1)**
Given the system:
\[
\begin{aligned}
x &= -2y, \\
x - y &= 9.
\end{aligned}
\]
Step 1: Substitute \(x = -2y\) into the second equation:
\[
-2y - y = 9.
\]
Step 2: Combine like terms:
\[
-3y = 9.
\]
Step 3: Solve for \(y\):
\[
y = \frac{9}{-3} = -3.
\]
Step 4: Substitute \(y = -3\) into \(x = -2y\):
\[
x = -2(-3) = 6.
\]
Solution: \(\boxed{(6,\,-3)}\).
---
**3)**
Given the system:
\[
\begin{aligned}
-x - 3y &= -18, \\
y &= -4x + 6.
\end{aligned}
\]
Step 1: Substitute \(y = -4x + 6\) into the first equation:
\[
-x - 3(-4x + 6) = -18.
\]
Step 2: Distribute \(-3\):
\[
-x + 12x - 18 = -18.
\]
Step 3: Combine like terms:
\[
11x - 18 = -18.
\]
Step 4: Solve for \(x\):
\[
11x = 0 \quad \Rightarrow \quad x = 0.
\]
Step 5: Substitute \(x = 0\) into \(y = -4x + 6\):
\[
y = -4(0) + 6 = 6.
\]
Solution: \(\boxed{(0,\,6)}\).
---
**5)**
Given the system:
\[
\begin{aligned}
y &= -6, \\
3x - 6y &= 30.
\end{aligned}
\]
Step 1: Substitute \(y = -6\) into the second equation:
\[
3x - 6(-6) = 30.
\]
Step 2: Simplify:
\[
3x + 36 = 30.
\]
Step 3: Solve for \(x\):
\[
3x = 30 - 36 = -6 \quad \Rightarrow \quad x = \frac{-6}{3} = -2.
\]
Solution: \(\boxed{(-2,\,-6)}\).
---
**2)**
Given the system:
\[
\begin{aligned}
y &= x - 4, \\
-4x - 6y &= -16.
\end{aligned}
\]
Step 1: Substitute \(y = x - 4\) into the second equation:
\[
-4x - 6(x - 4) = -16.
\]
Step 2: Distribute and simplify:
\[
-4x - 6x + 24 = -16 \quad \Rightarrow \quad -10x + 24 = -16.
\]
Step 3: Solve for \(x\):
\[
-10x = -16 - 24 = -40 \quad \Rightarrow \quad x = \frac{-40}{-10} = 4.
\]
Step 4: Substitute \(x = 4\) into \(y = x - 4\):
\[
y = 4 - 4 = 0.
\]
Solution: \(\boxed{(4,\,0)}\).
---
**4)**
Given the system:
\[
\begin{aligned}
-2x + 2y &= 6, \\
4x + 2y &= -5.
\end{aligned}
\]
Step 1: Subtract the first equation from the second to eliminate \(y\):
\[
(4x + 2y) - (-2x + 2y) = -5 - 6.
\]
Step 2: Simplify:
\[
4x + 2y + 2x - 2y = -11 \quad \Rightarrow \quad 6x = -11.
\]
Step 3: Solve for \(x\):
\[
x = -\frac{11}{6}.
\]
Step 4: Substitute \(x = -\frac{11}{6}\) into the first equation:
\[
-2\left(-\frac{11}{6}\right) + 2y = 6.
\]
Step 5: Simplify:
\[
\frac{22}{6} + 2y = 6 \quad \Rightarrow \quad \frac{11}{3} + 2y = 6.
\]
Step 6: Solve for \(y\):
\[
2y = 6 - \frac{11}{3} = \frac{18}{3} - \frac{11}{3} = \frac{7}{3} \quad \Rightarrow \quad y = \frac{7}{6}.
\]
Solution: \(\boxed{\left(-\frac{11}{6},\,\frac{7}{6}\right)}\).
---
**G)**
Given the system:
\[
\begin{aligned}
-2x - y &= -5, \\
x - 8y &= -23.
\end{aligned}
\]
Step 1: Solve the first equation for \(y\). Begin by rewriting:
\[
-2x - y = -5 \quad \Rightarrow \quad y = -2x + 5.
\]
Step 2: Substitute \(y = 5 - 2x\) into the second equation:
\[
x - 8(5 - 2x) = -23.
\]
Step 3: Distribute \(-8\):
\[
x - 40 + 16x = -23.
\]
Step 4: Combine like terms:
\[
17x - 40 = -23.
\]
Step 5: Solve for \(x\):
\[
17x = -23 + 40 = 17 \quad \Rightarrow \quad x = 1.
\]
Step 6: Substitute \(x = 1\) back into \(y = 5 - 2x\):
\[
y = 5 - 2(1) = 3.
\]
Solution: \(\boxed{(1,\,3)}\).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Mind Expander
To solve these systems, you first isolate one variable and then substitute it into the other equation. For example, in the first system, substitute \( x = -2y \) into \( x - y = 9 \) to find the value of \( y \). Then use that value to determine \( x \). Repeat this process for each system, and don’t forget to check your solutions in both original equations to ensure they satisfy the systems! When approaching these problems, common mistakes include forgetting to distribute negatives or misplacing values when substituting. To avoid errors, double-check each step and make sure everything is neatly organized. Using a separate piece of paper to jot down intermediate steps can really help keep track of your calculations and prevent confusion!
