Questions: Solve the system of equation (only real solutions) by the elimination method. Check your solutions. x^2 + y^2 = 53 x^2 - y^2 = 45

Solve the system of equation (only real solutions) by the elimination method. Check your solutions. x^2 + y^2 = 53 x^2 - y^2 = 45
Transcript text: Solve the system of equation (only real solutions) by the elimination method. Check your solutions. \[ \left\{\begin{array}{l} x^{2}+y^{2}=53 \\ x^{2}-y^{2}=45 \end{array}\right. \]
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Solution

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Solution Steps

To solve the given system of equations using the elimination method, we can subtract the second equation from the first. This will eliminate \( y^2 \) and allow us to solve for \( x^2 \). Once we have \( x^2 \), we can substitute it back into one of the original equations to find \( y^2 \). Finally, we take the square roots to find the values of \( x \) and \( y \).

To solve the given system of equations using the elimination method, we will follow these steps:

Step 1: Write Down the System of Equations

The given system of equations is: \[ \begin{align_} x^2 + y^2 &= 53 \quad \text{(Equation 1)} \\ x^2 - y^2 &= 45 \quad \text{(Equation 2)} \end{align_} \]

Step 2: Eliminate One Variable

To eliminate one of the variables, we can add the two equations together. This will eliminate \(y^2\).

\[ (x^2 + y^2) + (x^2 - y^2) = 53 + 45 \]

Simplifying, we get: \[ 2x^2 = 98 \]

Step 3: Solve for \(x^2\)

Divide both sides by 2 to solve for \(x^2\): \[ x^2 = \frac{98}{2} = 49 \]

Step 4: Solve for \(x\)

Take the square root of both sides to find \(x\): \[ x = \pm \sqrt{49} = \pm 7 \]

Step 5: Substitute Back to Find \(y\)

Substitute \(x = 7\) and \(x = -7\) back into one of the original equations to find \(y\). We will use Equation 1: \(x^2 + y^2 = 53\).

Case 1: \(x = 7\)

\[ 7^2 + y^2 = 53 \] \[ 49 + y^2 = 53 \] \[ y^2 = 53 - 49 = 4 \] \[ y = \pm \sqrt{4} = \pm 2 \]

Thus, the solutions for \(x = 7\) are \((7, 2)\) and \((7, -2)\).

Case 2: \(x = -7\)

\[ (-7)^2 + y^2 = 53 \] \[ 49 + y^2 = 53 \] \[ y^2 = 53 - 49 = 4 \] \[ y = \pm \sqrt{4} = \pm 2 \]

Thus, the solutions for \(x = -7\) are \((-7, 2)\) and \((-7, -2)\).

Step 6: Verify the Solutions

We need to verify that these solutions satisfy both original equations.

For \((7, 2)\):

  • Equation 1: \(7^2 + 2^2 = 49 + 4 = 53\) (True)
  • Equation 2: \(7^2 - 2^2 = 49 - 4 = 45\) (True)

For \((7, -2)\):

  • Equation 1: \(7^2 + (-2)^2 = 49 + 4 = 53\) (True)
  • Equation 2: \(7^2 - (-2)^2 = 49 - 4 = 45\) (True)

For \((-7, 2)\):

  • Equation 1: \((-7)^2 + 2^2 = 49 + 4 = 53\) (True)
  • Equation 2: \((-7)^2 - 2^2 = 49 - 4 = 45\) (True)

For \((-7, -2)\):

  • Equation 1: \((-7)^2 + (-2)^2 = 49 + 4 = 53\) (True)
  • Equation 2: \((-7)^2 - (-2)^2 = 49 - 4 = 45\) (True)

Final Answer

The solution set is: \[ \boxed{(7, 2), (7, -2), (-7, 2), (-7, -2)} \]

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