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

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:

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_}

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: $$2x^2 = 98$$

Divide both sides by 2 to solve for $x^2$: $$x^2 = \frac{98}{2} = 49$$

Take the square root of both sides to find $x$: $$x = \pm \sqrt{49} = \pm 7$$

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

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

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

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

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

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