Answer
Solutions:
1. \( x = -2 \), \( y = -3 \)
2. \( x = 0 \), \( y = -5 \)
3. \( x = -3 \), \( y = 3 \)
Solution
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}5x-2y=10\\y=-3x-5\end{array}\right.\)
- step1: Substitute the value of \(y:\)
\(5x-2\left(-3x-5\right)=10\)
- step2: Simplify:
\(11x+10=10\)
- step3: Move the constant to the right side:
\(11x=10-10\)
- step4: Subtract the terms:
\(11x=0\)
- step5: Rewrite the expression:
\(x=0\)
- step6: Substitute the value of \(x:\)
\(y=-3\times 0-5\)
- step7: Simplify:
\(y=-5\)
- step8: Calculate:
\(\left\{ \begin{array}{l}x=0\\y=-5\end{array}\right.\)
- step9: Check the solution:
\(\left\{ \begin{array}{l}x=0\\y=-5\end{array}\right.\)
- step10: Rewrite:
\(\left(x,y\right) = \left(0,-5\right)\)
Solve the system of equations \( 4 x+4 y=0;y=-5 x-12 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}4x+4y=0\\y=-5x-12\end{array}\right.\)
- step1: Substitute the value of \(y:\)
\(4x+4\left(-5x-12\right)=0\)
- step2: Simplify:
\(-16x-48=0\)
- step3: Move the constant to the right side:
\(-16x=0+48\)
- step4: Remove 0:
\(-16x=48\)
- step5: Change the signs:
\(16x=-48\)
- step6: Divide both sides:
\(\frac{16x}{16}=\frac{-48}{16}\)
- step7: Divide the numbers:
\(x=-3\)
- step8: Substitute the value of \(x:\)
\(y=-5\left(-3\right)-12\)
- step9: Simplify:
\(y=3\)
- step10: Calculate:
\(\left\{ \begin{array}{l}x=-3\\y=3\end{array}\right.\)
- step11: Check the solution:
\(\left\{ \begin{array}{l}x=-3\\y=3\end{array}\right.\)
- step12: Rewrite:
\(\left(x,y\right) = \left(-3,3\right)\)
Solve the system of equations \( y=-3 x-9;5 x+2 y=-16 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}y=-3x-9\\5x+2y=-16\end{array}\right.\)
- step1: Substitute the value of \(y:\)
\(5x+2\left(-3x-9\right)=-16\)
- step2: Simplify:
\(-x-18=-16\)
- step3: Move the constant to the right side:
\(-x=-16+18\)
- step4: Add the numbers:
\(-x=2\)
- step5: Change the signs:
\(x=-2\)
- step6: Substitute the value of \(x:\)
\(y=-3\left(-2\right)-9\)
- step7: Simplify:
\(y=-3\)
- step8: Calculate:
\(\left\{ \begin{array}{l}x=-2\\y=-3\end{array}\right.\)
- step9: Check the solution:
\(\left\{ \begin{array}{l}x=-2\\y=-3\end{array}\right.\)
- step10: Rewrite:
\(\left(x,y\right) = \left(-2,-3\right)\)
Here are the solutions for each of the systems of equations using the substitution method:
### 4.
Given the equations:
\[
\begin{array}{l}
y = -3x - 9 \\
5x + 2y = -16
\end{array}
\]
**Substituting \(y\) into the second equation:**
\[
5x + 2(-3x - 9) = -16
\]
\[
5x - 6x - 18 = -16
\]
\[
-x - 18 = -16
\]
\[
-x = 2 \implies x = -2
\]
**Now substituting \(x\) back to find \(y\):**
\[
y = -3(-2) - 9 = 6 - 9 = -3
\]
**Solution:**
\[
(x, y) = (-2, -3)
\]
### 5.
Given the equations:
\[
\begin{array}{l}
5x - 2y = 10 \\
y = -3x - 5
\end{array}
\]
**Substituting \(y\) into the first equation:**
\[
5x - 2(-3x - 5) = 10
\]
\[
5x + 6x + 10 = 10
\]
\[
11x + 10 = 10
\]
\[
11x = 0 \implies x = 0
\]
**Now substituting \(x\) back to find \(y\):**
\[
y = -3(0) - 5 = -5
\]
**Solution:**
\[
(x, y) = (0, -5)
\]
### 6.
Given the equations:
\[
\begin{array}{l}
4x + 4y = 0 \\
y = -5x - 12
\end{array}
\]
**Substituting \(y\) into the first equation:**
\[
4x + 4(-5x - 12) = 0
\]
\[
4x - 20x - 48 = 0
\]
\[
-16x - 48 = 0
\]
\[
-16x = 48 \implies x = -3
\]
**Now substituting \(x\) back to find \(y\):**
\[
y = -5(-3) - 12 = 15 - 12 = 3
\]
**Solution:**
\[
(x, y) = (-3, 3)
\]
### Summary of Solutions:
- For question 4: \((x, y) = (-2, -3)\)
- For question 5: \((x, y) = (0, -5)\)
- For question 6: \((x, y) = (-3, 3)\)
Reviewed and approved by the UpStudy tutoring team
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