Solve x ^ 2 + 2y ^ 2 = 3; x - 3y = 2 | 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

$$x^{2}+2y^{2}=3; x-3y=2$$

Solve for x, y

$x=\frac{19}{11}\approx 1.727272727\text{, }y=-\frac{1}{11}\approx -0.090909091$
$x=-1\text{, }y=-1$

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.

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

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

$$x-3y=2$$

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

$$x=3y+2$$

Substitute $3y+2$ for $x$ in the other equation, $2y^{2}+x^{2}=3$.

$$2y^{2}+\left(3y+2\right)^{2}=3$$

Square $3y+2$.

$$2y^{2}+9y^{2}+12y+4=3$$

Add $2y^{2}$ to $9y^{2}$.

$$11y^{2}+12y+4=3$$

Subtract $3$ from both sides of the equation.

$$11y^{2}+12y+1=0$$

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

$$y=\frac{-12±\sqrt{12^{2}-4\times 11}}{2\times 11}$$

Square $1\times 2\times 2\times 3$.

$$y=\frac{-12±\sqrt{144-4\times 11}}{2\times 11}$$

Multiply $-4$ times $2+1\times 3^{2}$.

$$y=\frac{-12±\sqrt{144-44}}{2\times 11}$$

Add $144$ to $-44$.

$$y=\frac{-12±\sqrt{100}}{2\times 11}$$

Take the square root of $100$.

$$y=\frac{-12±10}{2\times 11}$$

Multiply $2$ times $2+1\times 3^{2}$.

$$y=\frac{-12±10}{22}$$

Now solve the equation $y=\frac{-12±10}{22}$ when $±$ is plus. Add $-12$ to $10$.

$$y=-\frac{2}{22}$$

Reduce the fraction $\frac{-2}{22}$ to lowest terms by extracting and canceling out $2$.

$$y=-\frac{1}{11}$$

Now solve the equation $y=\frac{-12±10}{22}$ when $±$ is minus. Subtract $10$ from $-12$.

$$y=-\frac{22}{22}$$

Divide $-22$ by $22$.

$$y=-1$$

There are two solutions for $y$: $-\frac{1}{11}$ and $-1$. Substitute $-\frac{1}{11}$ for $y$ in the equation $x=3y+2$ to find the corresponding solution for $x$ that satisfies both equations.

$$x=3\left(-\frac{1}{11}\right)+2$$

Multiply $3$ times $-\frac{1}{11}$.

$$x=-\frac{3}{11}+2$$

Add $-\frac{1}{11}\times 3$ to $2$.

$$x=\frac{19}{11}$$

Now substitute $-1$ for $y$ in the equation $x=3y+2$ and solve to find the corresponding solution for $x$ that satisfies both equations.

$$x=3\left(-1\right)+2$$

Multiply $3$ times $-1$.

$$x=-3+2$$

Add $-3$ to $2$.

$$x=-1$$

The system is now solved.

$$x=\frac{19}{11},y=-\frac{1}{11}\text{ or }x=-1,y=-1$$