Math / Algebra

QuestionUse the elimination method to find all solutions of the system. Write the solution(5) as a list of ordered pairs. $\left\{\begin{array}{l} x^{2}-4 y=32 \\ x^{2}+2 y=8 \end{array}\right.$

Studdy Solution

STEP 1

1. We are given a system of two equations with two variables, $x$ and $y$ .
2. The elimination method involves eliminating one variable by adding or subtracting the equations.
3. The goal is to find all solutions as ordered pairs $(x, y)$ .

STEP 2

1. Align the equations for elimination.
2. Eliminate one variable by subtracting the equations.
3. Solve for the remaining variable.
4. Substitute back to find the other variable.
5. Write the solutions as ordered pairs.

STEP 3

Align the given system of equations for elimination:
$\begin{array}{l} x^{2}-4y=32 \\ x^{2}+2y=8 \end{array}$

STEP 4

Subtract the second equation from the first to eliminate $x^2$ :
$(x^{2} - 4y) - (x^{2} + 2y) = 32 - 8$
Simplify:
$x^2 - 4y - x^2 - 2y = 24$
$-6y = 24$

STEP 5

Solve for $y$ :
$-6y = 24$
Divide both sides by $-6$ :
$y = -4$

STEP 6

Substitute $y = -4$ back into one of the original equations to solve for $x^2$ . Let's use the second equation:
$x^{2} + 2(-4) = 8$
Simplify:
$x^{2} - 8 = 8$
Add 8 to both sides:
$x^{2} = 16$
Take the square root of both sides:
$x = 4 \quad \text{or} \quad x = -4$

STEP 7

Write the solutions as ordered pairs. Since $y = -4$ , the solutions are:
$(4, -4) \quad \text{and} \quad (-4, -4)$
The solutions to the system are:
$\boxed{(4, -4), (-4, -4)}$

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QuestionUse the elimination method to find all solutions of the system. Write the solution(5) as a list of ordered pairs. $\left\{\begin{array}{l} x^{2}-4 y=32 \\ x^{2}+2 y=8 \end{array}\right.$

Studdy Solution

STEP 1

1. We are given a system of two equations with two variables, $x$ and $y$ .
2. The elimination method involves eliminating one variable by adding or subtracting the equations.
3. The goal is to find all solutions as ordered pairs $(x, y)$ .

STEP 2

1. Align the equations for elimination.
2. Eliminate one variable by subtracting the equations.
3. Solve for the remaining variable.
4. Substitute back to find the other variable.
5. Write the solutions as ordered pairs.

STEP 3

Align the given system of equations for elimination:
$\begin{array}{l} x^{2}-4y=32 \\ x^{2}+2y=8 \end{array}$

STEP 4

Subtract the second equation from the first to eliminate $x^2$ :
$(x^{2} - 4y) - (x^{2} + 2y) = 32 - 8$
Simplify:
$x^2 - 4y - x^2 - 2y = 24$
$-6y = 24$

STEP 5

Solve for $y$ :
$-6y = 24$
Divide both sides by $-6$ :
$y = -4$

STEP 6

Substitute $y = -4$ back into one of the original equations to solve for $x^2$ . Let's use the second equation:
$x^{2} + 2(-4) = 8$
Simplify:
$x^{2} - 8 = 8$
Add 8 to both sides:
$x^{2} = 16$
Take the square root of both sides:
$x = 4 \quad \text{or} \quad x = -4$

STEP 7

Write the solutions as ordered pairs. Since $y = -4$ , the solutions are:
$(4, -4) \quad \text{and} \quad (-4, -4)$
The solutions to the system are:
$\boxed{(4, -4), (-4, -4)}$

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QuestionUse the elimination method to find all solutions of the system. Write the solution(5) as a list of ordered pairs. {x2−4y=32x2+2y=8\left\{\begin{array}{l} x^{2}-4 y=32 \\ x^{2}+2 y=8 \end{array}\right.{x2−4y=32x2+2y=8​

Studdy Solution

STEP 1

1. We are given a system of two equations with two variables, x x x and y y y.2. The elimination method involves eliminating one variable by adding or subtracting the equations.3. The goal is to find all solutions as ordered pairs (x,y)(x, y)(x,y).

STEP 2

1. Align the equations for elimination.2. Eliminate one variable by subtracting the equations.3. Solve for the remaining variable.4. Substitute back to find the other variable.5. Write the solutions as ordered pairs.

STEP 3

Align the given system of equations for elimination:x2−4y=32x2+2y=8\begin{array}{l} x^{2}-4y=32 \\ x^{2}+2y=8 \end{array}x2−4y=32x2+2y=8​

STEP 4

Subtract the second equation from the first to eliminate x2 x^2 x2:(x2−4y)−(x2+2y)=32−8(x^{2} - 4y) - (x^{2} + 2y) = 32 - 8(x2−4y)−(x2+2y)=32−8Simplify:x2−4y−x2−2y=24x^2 - 4y - x^2 - 2y = 24x2−4y−x2−2y=24−6y=24-6y = 24−6y=24

STEP 5

Solve for y y y:−6y=24-6y = 24−6y=24Divide both sides by −6-6−6:y=−4y = -4y=−4

STEP 6

STEP 7

Write the solutions as ordered pairs. Since y=−4 y = -4 y=−4, the solutions are:(4,−4)and(−4,−4)(4, -4) \quad \text{and} \quad (-4, -4)(4,−4)and(−4,−4) The solutions to the system are:(4,−4),(−4,−4)\boxed{(4, -4), (-4, -4)}(4,−4),(−4,−4)​

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QuestionUse the elimination method to find all solutions of the system. Write the solution(5) as a list of ordered pairs. $\left\{\begin{array}{l} x^{2}-4 y=32 \\ x^{2}+2 y=8 \end{array}\right.$

1. We are given a system of two equations with two variables, $x$ and $y$ .
2. The elimination method involves eliminating one variable by adding or subtracting the equations.
3. The goal is to find all solutions as ordered pairs $(x, y)$ .

1. Align the equations for elimination.
2. Eliminate one variable by subtracting the equations.
3. Solve for the remaining variable.
4. Substitute back to find the other variable.
5. Write the solutions as ordered pairs.

Align the given system of equations for elimination:
$\begin{array}{l} x^{2}-4y=32 \\ x^{2}+2y=8 \end{array}$

Subtract the second equation from the first to eliminate $x^2$ :
$(x^{2} - 4y) - (x^{2} + 2y) = 32 - 8$
Simplify:
$x^2 - 4y - x^2 - 2y = 24$
$-6y = 24$

Solve for $y$ :
$-6y = 24$
Divide both sides by $-6$ :
$y = -4$

Write the solutions as ordered pairs. Since $y = -4$ , the solutions are:
$(4, -4) \quad \text{and} \quad (-4, -4)$
The solutions to the system are:
$\boxed{(4, -4), (-4, -4)}$