Solve (x)^(2)-(y)^(2)=16,2x+3y=19 | 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

$${ x }^{ 2 } - { y }^{ 2 } =16,2x+3y=19$$

Solve for x, y

$x=5\text{, }y=3$
$x=-\frac{101}{5}=-20.2\text{, }y=\frac{99}{5}=19.8$

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.

$$2x+3y=19,-y^{2}+x^{2}=16$$

Solve $2x+3y=19$ for $x$ by isolating $x$ on the left hand side of the equal sign.

$$2x+3y=19$$

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

$$2x=-3y+19$$

Divide both sides by $2$.

$$x=-\frac{3}{2}y+\frac{19}{2}$$

Substitute $-\frac{3}{2}y+\frac{19}{2}$ for $x$ in the other equation, $-y^{2}+x^{2}=16$.

$$-y^{2}+\left(-\frac{3}{2}y+\frac{19}{2}\right)^{2}=16$$

Square $-\frac{3}{2}y+\frac{19}{2}$.

$$-y^{2}+\frac{9}{4}y^{2}-\frac{57}{2}y+\frac{361}{4}=16$$

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

$$\frac{5}{4}y^{2}-\frac{57}{2}y+\frac{361}{4}=16$$

Subtract $16$ from both sides of the equation.

$$\frac{5}{4}y^{2}-\frac{57}{2}y+\frac{297}{4}=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-1+1\left(-\frac{3}{2}\right)^{2}$ for $a$, $1\times \frac{19}{2}\left(-\frac{3}{2}\right)\times 2$ for $b$, and $\frac{297}{4}$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$y=\frac{-\left(-\frac{57}{2}\right)±\sqrt{\left(-\frac{57}{2}\right)^{2}-4\times \frac{5}{4}\times \frac{297}{4}}}{2\times \frac{5}{4}}$$

Square $1\times \frac{19}{2}\left(-\frac{3}{2}\right)\times 2$.

$$y=\frac{-\left(-\frac{57}{2}\right)±\sqrt{\frac{3249}{4}-4\times \frac{5}{4}\times \frac{297}{4}}}{2\times \frac{5}{4}}$$

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

$$y=\frac{-\left(-\frac{57}{2}\right)±\sqrt{\frac{3249}{4}-5\times \frac{297}{4}}}{2\times \frac{5}{4}}$$

Multiply $-5$ times $\frac{297}{4}$.

$$y=\frac{-\left(-\frac{57}{2}\right)±\sqrt{\frac{3249-1485}{4}}}{2\times \frac{5}{4}}$$

Add $\frac{3249}{4}$ to $-\frac{1485}{4}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{57}{2}\right)±\sqrt{441}}{2\times \frac{5}{4}}$$

Take the square root of $441$.

$$y=\frac{-\left(-\frac{57}{2}\right)±21}{2\times \frac{5}{4}}$$

The opposite of $1\times \frac{19}{2}\left(-\frac{3}{2}\right)\times 2$ is $\frac{57}{2}$.

$$y=\frac{\frac{57}{2}±21}{2\times \frac{5}{4}}$$

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

$$y=\frac{\frac{57}{2}±21}{\frac{5}{2}}$$

Now solve the equation $y=\frac{\frac{57}{2}±21}{\frac{5}{2}}$ when $±$ is plus. Add $\frac{57}{2}$ to $21$.

$$y=\frac{\frac{99}{2}}{\frac{5}{2}}$$

Divide $\frac{99}{2}$ by $\frac{5}{2}$ by multiplying $\frac{99}{2}$ by the reciprocal of $\frac{5}{2}$.

$$y=\frac{99}{5}$$

Now solve the equation $y=\frac{\frac{57}{2}±21}{\frac{5}{2}}$ when $±$ is minus. Subtract $21$ from $\frac{57}{2}$.

$$y=\frac{\frac{15}{2}}{\frac{5}{2}}$$

Divide $\frac{15}{2}$ by $\frac{5}{2}$ by multiplying $\frac{15}{2}$ by the reciprocal of $\frac{5}{2}$.

$$y=3$$

There are two solutions for $y$: $\frac{99}{5}$ and $3$. Substitute $\frac{99}{5}$ for $y$ in the equation $x=-\frac{3}{2}y+\frac{19}{2}$ to find the corresponding solution for $x$ that satisfies both equations.

$$x=-\frac{3}{2}\times \frac{99}{5}+\frac{19}{2}$$

Multiply $-\frac{3}{2}$ times $\frac{99}{5}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{297}{10}+\frac{19}{2}$$

Add $-\frac{3}{2}\times \frac{99}{5}$ to $\frac{19}{2}$.

$$x=-\frac{101}{5}$$

Now substitute $3$ for $y$ in the equation $x=-\frac{3}{2}y+\frac{19}{2}$ and solve to find the corresponding solution for $x$ that satisfies both equations.

$$x=-\frac{3}{2}\times 3+\frac{19}{2}$$

Multiply $-\frac{3}{2}$ times $3$.

$$x=\frac{-9+19}{2}$$

Add $-\frac{3}{2}\times 3$ to $\frac{19}{2}$.

$$x=5$$

The system is now solved.

$$x=-\frac{101}{5},y=\frac{99}{5}\text{ or }x=5,y=3$$