Solve x ^ 2 - y ^ 2 = 106; 2x - 3y = 8 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$x^{2}-y^{2}=106; 2x-3y=8$$

Solve for x, y

$x=\frac{-9\sqrt{66}-16}{5}\approx -17.823269128\text{, }y=\frac{-6\sqrt{66}-24}{5}\approx -14.548846086$
$x=\frac{9\sqrt{66}-16}{5}\approx 11.423269128\text{, }y=\frac{6\sqrt{66}-24}{5}\approx 4.948846086$

Steps Using Substitution and Quadratic Formula

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$2x-3y=8,-y^{2}+x^{2}=106$$

Solve $2x-3y=8$ for $x$ by isolating $x$ on the left hand side of the equal sign.

$$2x-3y=8$$

Subtract $-3y$ from both sides of the equation.

$$2x=3y+8$$

Divide both sides by $2$.

$$x=\frac{3}{2}y+4$$

Substitute $\frac{3}{2}y+4$ for $x$ in the other equation, $-y^{2}+x^{2}=106$.

$$-y^{2}+\left(\frac{3}{2}y+4\right)^{2}=106$$

Square $\frac{3}{2}y+4$.

$$-y^{2}+\frac{9}{4}y^{2}+12y+16=106$$

Add $-y^{2}$ to $\frac{9}{4}y^{2}$.

$$\frac{5}{4}y^{2}+12y+16=106$$

Subtract $106$ from both sides of the equation.

$$\frac{5}{4}y^{2}+12y-90=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-1+1\times \left(\frac{3}{2}\right)^{2}$ for $a$, $1\times 4\times \frac{3}{2}\times 2$ for $b$, and $-90$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$y=\frac{-12±\sqrt{12^{2}-4\times \frac{5}{4}\left(-90\right)}}{2\times \frac{5}{4}}$$

Square $1\times 4\times \frac{3}{2}\times 2$.

$$y=\frac{-12±\sqrt{144-4\times \frac{5}{4}\left(-90\right)}}{2\times \frac{5}{4}}$$

Multiply $-4$ times $-1+1\times \left(\frac{3}{2}\right)^{2}$.

$$y=\frac{-12±\sqrt{144-5\left(-90\right)}}{2\times \frac{5}{4}}$$

Multiply $-5$ times $-90$.

$$y=\frac{-12±\sqrt{144+450}}{2\times \frac{5}{4}}$$

Add $144$ to $450$.

$$y=\frac{-12±\sqrt{594}}{2\times \frac{5}{4}}$$

Take the square root of $594$.

$$y=\frac{-12±3\sqrt{66}}{2\times \frac{5}{4}}$$

Multiply $2$ times $-1+1\times \left(\frac{3}{2}\right)^{2}$.

$$y=\frac{-12±3\sqrt{66}}{\frac{5}{2}}$$

Now solve the equation $y=\frac{-12±3\sqrt{66}}{\frac{5}{2}}$ when $±$ is plus. Add $-12$ to $3\sqrt{66}$.

$$y=\frac{3\sqrt{66}-12}{\frac{5}{2}}$$

Divide $-12+3\sqrt{66}$ by $\frac{5}{2}$ by multiplying $-12+3\sqrt{66}$ by the reciprocal of $\frac{5}{2}$.

$$y=\frac{6\sqrt{66}-24}{5}$$

Now solve the equation $y=\frac{-12±3\sqrt{66}}{\frac{5}{2}}$ when $±$ is minus. Subtract $3\sqrt{66}$ from $-12$.

$$y=\frac{-3\sqrt{66}-12}{\frac{5}{2}}$$

Divide $-12-3\sqrt{66}$ by $\frac{5}{2}$ by multiplying $-12-3\sqrt{66}$ by the reciprocal of $\frac{5}{2}$.

$$y=\frac{-6\sqrt{66}-24}{5}$$

There are two solutions for $y$: $\frac{-24+6\sqrt{66}}{5}$ and $\frac{-24-6\sqrt{66}}{5}$. Substitute $\frac{-24+6\sqrt{66}}{5}$ for $y$ in the equation $x=\frac{3}{2}y+4$ to find the corresponding solution for $x$ that satisfies both equations.

$$x=\frac{3}{2}\times \frac{6\sqrt{66}-24}{5}+4$$

Multiply $\frac{3}{2}$ times $\frac{-24+6\sqrt{66}}{5}$.

$$x=\frac{3\times \frac{6\sqrt{66}-24}{5}}{2}+4$$

Now substitute $\frac{-24-6\sqrt{66}}{5}$ for $y$ in the equation $x=\frac{3}{2}y+4$ and solve to find the corresponding solution for $x$ that satisfies both equations.

$$x=\frac{3}{2}\times \frac{-6\sqrt{66}-24}{5}+4$$

Multiply $\frac{3}{2}$ times $\frac{-24-6\sqrt{66}}{5}$.

$$x=\frac{3\times \frac{-6\sqrt{66}-24}{5}}{2}+4$$

The system is now solved.

$$x=\frac{3\times \frac{6\sqrt{66}-24}{5}}{2}+4,y=\frac{6\sqrt{66}-24}{5}\text{ or }x=\frac{3\times \frac{-6\sqrt{66}-24}{5}}{2}+4,y=\frac{-6\sqrt{66}-24}{5}$$