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Use elimination to solve each system of equations:

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Ask by Mullins Estrada. in the United States

Mar 18,2025

Question

Try it: Use elimination to solve each system of equations: $$ \begin{array}{l|l} y=-x+4 & y=-x^{2}+4 x+2 \ y=x^{2}-2 & y=2-x \end{array} $$

UpStudy AI · Accepted Answer

Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}y=x^{2}-2\y=2-x\end{array}\right.$$
- step1: Substitute the value of $$y:$$
  $$x^{2}-2=2-x$$
- step2: Move the expression to the left side:
  $$x^{2}-2-\left(2-x\right)=0$$
- step3: Subtract the terms:
  $$x^{2}-4+x=0$$
- step4: Rewrite in standard form:
  $$x^{2}+x-4=0$$
- step5: Solve using the quadratic formula:
  $$x=\frac{-1\pm \sqrt{1^{2}-4\left(-4\right)}}{2}$$
- step6: Simplify the expression:
  $$x=\frac{-1\pm \sqrt{17}}{2}$$
- step7: Separate into possible cases:
  $$\begin{align}&x=\frac{-1+\sqrt{17}}{2}\&x=\frac{-1-\sqrt{17}}{2}\end{align}$$
- step8: Rewrite the fraction:
  $$\begin{align}&x=\frac{-1+\sqrt{17}}{2}\&x=-\frac{1+\sqrt{17}}{2}\end{align}$$
- step9: Evaluate the logic:
  $$x=\frac{-1+\sqrt{17}}{2}\cup x=-\frac{1+\sqrt{17}}{2}$$
- step10: Rearrange the terms:
  $$\left\{ \begin{array}{l}x=\frac{-1+\sqrt{17}}{2}\y=x^{2}-2\end{array}\right.\cup \left\{ \begin{array}{l}x=-\frac{1+\sqrt{17}}{2}\y=x^{2}-2\end{array}\right.$$
- step11: Calculate:
  $$\left\{ \begin{array}{l}x=\frac{-1+\sqrt{17}}{2}\y=\frac{5-\sqrt{17}}{2}\end{array}\right.\cup \left\{ \begin{array}{l}x=-\frac{1+\sqrt{17}}{2}\y=\frac{5+\sqrt{17}}{2}\end{array}\right.$$
- step12: Calculate:
  $$\left\{ \begin{array}{l}x=-\frac{1+\sqrt{17}}{2}\y=\frac{5+\sqrt{17}}{2}\end{array}\right.\cup \left\{ \begin{array}{l}x=\frac{-1+\sqrt{17}}{2}\y=\frac{5-\sqrt{17}}{2}\end{array}\right.$$
- step13: Check the solution:
  $$\left\{ \begin{array}{l}x=-\frac{1+\sqrt{17}}{2}\y=\frac{5+\sqrt{17}}{2}\end{array}\right.\cup \left\{ \begin{array}{l}x=\frac{-1+\sqrt{17}}{2}\y=\frac{5-\sqrt{17}}{2}\end{array}\right.$$
- step14: Rewrite:
  $$\left(x,y\right) = \left(-\frac{1+\sqrt{17}}{2},\frac{5+\sqrt{17}}{2}\right)\cup \left(x,y\right) = \left(\frac{-1+\sqrt{17}}{2},\frac{5-\sqrt{17}}{2}\right)$$
Solve the system of equations $$ y=-x+4; y=-x^{2}+4 x+2 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}y=-x+4\y=-x^{2}+4x+2\end{array}\right.$$
- step1: Substitute the value of $$y:$$
  $$-x+4=-x^{2}+4x+2$$
- step2: Move the expression to the left side:
  $$-x+4-\left(-x^{2}+4x+2\right)=0$$
- step3: Calculate:
  $$-5x+2+x^{2}=0$$
- step4: Rewrite in standard form:
  $$x^{2}-5x+2=0$$
- step5: Solve using the quadratic formula:
  $$x=\frac{5\pm \sqrt{\left(-5\right)^{2}-4\times 2}}{2}$$
- step6: Simplify the expression:
  $$x=\frac{5\pm \sqrt{17}}{2}$$
- step7: Separate into possible cases:
  $$\begin{align}&x=\frac{5+\sqrt{17}}{2}\&x=\frac{5-\sqrt{17}}{2}\end{align}$$
- step8: Evaluate the logic:
  $$x=\frac{5+\sqrt{17}}{2}\cup x=\frac{5-\sqrt{17}}{2}$$
- step9: Rearrange the terms:
  $$\left\{ \begin{array}{l}x=\frac{5+\sqrt{17}}{2}\y=-x+4\end{array}\right.\cup \left\{ \begin{array}{l}x=\frac{5-\sqrt{17}}{2}\y=-x+4\end{array}\right.$$
- step10: Calculate:
  $$\left\{ \begin{array}{l}x=\frac{5+\sqrt{17}}{2}\y=\frac{3-\sqrt{17}}{2}\end{array}\right.\cup \left\{ \begin{array}{l}x=\frac{5-\sqrt{17}}{2}\y=\frac{\sqrt{17}+3}{2}\end{array}\right.$$
- step11: Check the solution:
  $$\left\{ \begin{array}{l}x=\frac{5+\sqrt{17}}{2}\y=\frac{3-\sqrt{17}}{2}\end{array}\right.\cup \left\{ \begin{array}{l}x=\frac{5-\sqrt{17}}{2}\y=\frac{\sqrt{17}+3}{2}\end{array}\right.$$
- step12: Rewrite:
  $$\left(x,y\right) = \left(\frac{5+\sqrt{17}}{2},\frac{3-\sqrt{17}}{2}\right)\cup \left(x,y\right) = \left(\frac{5-\sqrt{17}}{2},\frac{\sqrt{17}+3}{2}\right)$$
Let's solve the given systems of equations using the elimination method step by step.

### System 1:
$$
\begin{align*}
1. & \quad y = -x + 4 \quad (1) \
2. & \quad y = -x^2 + 4x + 2 \quad (2)
\end{align*}
$$

**Step 1:** Substitute equation (1) into equation (2):
$$
-x + 4 = -x^2 + 4x + 2
$$

**Step 2:** Rearranging gives:
$$
-x^2 + 4x + 2 + x - 4 = 0 \
-x^2 + 5x - 2 = 0
$$

**Step 3:** Multiply through by -1:
$$
x^2 - 5x + 2 = 0
$$

**Step 4:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$:
$$
x = \frac{5 \pm \sqrt{25 - 8}}{2} = \frac{5 \pm \sqrt{17}}{2}
$$

**Step 5:** Substitute $$x$$ back into equation (1) to find $$y$$:
$$
y = -\left(\frac{5 + \sqrt{17}}{2}\right) + 4 = \frac{3 - \sqrt{17}}{2} \
y = -\left(\frac{5 - \sqrt{17}}{2}\right) + 4 = \frac{\sqrt{17} + 3}{2}
$$

**Final Solutions for System 1:**
$$
(x,y) = \left(\frac{5+\sqrt{17}}{2}, \frac{3-\sqrt{17}}{2}\right) \cup \left(x,y\right) = \left(\frac{5-\sqrt{17}}{2}, \frac{\sqrt{17}+3}{2}\right)
$$

---

### System 2:
$$
\begin{align*}
1. & \quad y = x^2 - 2 \quad (1) \
2. & \quad y = 2 - x \quad (2)
\end{align*}
$$

**Step 1:** Substitute equation (2) into equation (1):
$$
2 - x = x^2 - 2
$$

**Step 2:** Rearranging gives:
$$
x^2 + x - 4 = 0
$$

**Step 3:** Use the quadratic formula:
$$
x = \frac{-1 \pm \sqrt{1 + 16}}{2} = \frac{-1 \pm \sqrt{17}}{2}
$$

**Step 4:** Substitute $$x$$ back into equation (2) to find $$y$$:
$$
y = 2 - \left(-\frac{1 + \sqrt{17}}{2}\right) = \frac{5 + \sqrt{17}}{2} \
y = 2 - \left(\frac{-1 + \sqrt{17}}{2}\right) = \frac{5 - \sqrt{17}}{2}
$$

**Final Solutions for System 2:**
$$
(x,y) = \left(-\frac{1+\sqrt{17}}{2}, \frac{5+\sqrt{17}}{2}\right) \cup \left(x,y\right) = \left(\frac{-1+\sqrt{17}}{2}, \frac{5-\sqrt{17}}{2}\right)
$$

### Summary of Solutions:
- **System 1:** 
  $$
  (x,y) = \left(\frac{5+\sqrt{17}}{2}, \frac{3-\sqrt{17}}{2}\right) \cup \left(\frac{5-\sqrt{17}}{2}, \frac{\sqrt{17}+3}{2}\right)
  $$
  
- **System 2:** 
  $$
  (x,y) = \left(-\frac{1+\sqrt{17}}{2}, \frac{5+\sqrt{17}}{2}\right) \cup \left(\frac{-1+\sqrt{17}}{2}, \frac{5-\sqrt{17}}{2}\right)
  $$
The solutions for the systems are: - **System 1:** $$ \left(\frac{5+\sqrt{17}}{2}, \frac{3-\sqrt{17}}{2}\right) $$ and $$ \left(\frac{5-\sqrt{17}}{2}, \frac{\sqrt{17}+3}{2}\right) $$ - **System 2:** $$ \left(-\frac{1+\sqrt{17}}{2}, \frac{5+\sqrt{17}}{2}\right) $$ and $$ \left(\frac{-1+\sqrt{17}}{2}, \frac{5-\sqrt{17}}{2}\right) $$

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Use elimination to solve each system of equations:

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