Solve the System of Equations 3(y-4)-2(x-3)=-6 5x^2+2y^2-53=0

$3 (y - 4) - 2 (x - 3) = - 6$ $5 x^{2} + 2 y^{2} - 53 = 0$

Step 1

Solve for $y$ in $3 (y - 4) - 2 (x - 3) = - 6$ .

Simplify $3 (y - 4) - 2 (x - 3)$ .

Simplify each term.

Apply the distributive property.

$3 y + 3 \cdot - 4 - 2 (x - 3) = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Multiply $3$ by $- 4$ .

$3 y - 12 - 2 (x - 3) = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Apply the distributive property.

$3 y - 12 - 2 x - 2 \cdot - 3 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Multiply $- 2$ by $- 3$ .

$3 y - 12 - 2 x + 6 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

$3 y - 12 - 2 x + 6 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Add $- 12$ and $6$ .

$3 y - 2 x - 6 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

$3 y - 2 x - 6 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Move all terms not containing $y$ to the right side of the equation.

Add $2 x$ to both sides of the equation.

$3 y - 6 = - 6 + 2 x$

$5 x^{2} + 2 y^{2} - 53 = 0$

Add $6$ to both sides of the equation.

$3 y = - 6 + 2 x + 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Combine the opposite terms in $- 6 + 2 x + 6$ .

Add $- 6$ and $6$ .

$3 y = 2 x + 0$

$5 x^{2} + 2 y^{2} - 53 = 0$

Add $2 x$ and $0$ .

$3 y = 2 x$

$5 x^{2} + 2 y^{2} - 53 = 0$

$3 y = 2 x$

$5 x^{2} + 2 y^{2} - 53 = 0$

$3 y = 2 x$

$5 x^{2} + 2 y^{2} - 53 = 0$

Divide each term in $3 y = 2 x$ by $3$ and simplify.

Divide each term in $3 y = 2 x$ by $3$ .

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

$5 x^{2} + 2 y^{2} - 53 = 0$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

$5 x^{2} + 2 y^{2} - 53 = 0$

Divide $y$ by $1$ .

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

$5 x^{2} + 2 y^{2} - 53 = 0$

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

$5 x^{2} + 2 y^{2} - 53 = 0$

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

$5 x^{2} + 2 y^{2} - 53 = 0$

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

$5 x^{2} + 2 y^{2} - 53 = 0$

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

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 2

Replace all occurrences of $y$ with $\frac{2 x}{3}$ in each equation.

Replace all occurrences of $y$ in $5 x^{2} + 2 y^{2} - 53 = 0$ with $\frac{2 x}{3}$ .

$5 x^{2} + 2 {(\frac{2 x}{3})}^{2} - 53 = 0$

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

Simplify the left side.

Simplify $5 x^{2} + 2 {(\frac{2 x}{3})}^{2} - 53$ .

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{2 x}{3}$ .

$5 x^{2} + 2 (\frac{{(2 x)}^{2}}{3^{2}}) - 53 = 0$

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

Apply the product rule to $2 x$ .

$5 x^{2} + 2 (\frac{2^{2} x^{2}}{3^{2}}) - 53 = 0$

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

$5 x^{2} + 2 (\frac{2^{2} x^{2}}{3^{2}}) - 53 = 0$

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

Raise $2$ to the power of $2$ .

$5 x^{2} + 2 (\frac{4 x^{2}}{3^{2}}) - 53 = 0$

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

Raise $3$ to the power of $2$ .

$5 x^{2} + 2 (\frac{4 x^{2}}{9}) - 53 = 0$

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

Multiply $2 \frac{4 x^{2}}{9}$ .

Combine $2$ and $\frac{4 x^{2}}{9}$ .

$5 x^{2} + \frac{2 (4 x^{2})}{9} - 53 = 0$

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

Multiply $4$ by $2$ .

$5 x^{2} + \frac{8 x^{2}}{9} - 53 = 0$

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

$5 x^{2} + \frac{8 x^{2}}{9} - 53 = 0$

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

$5 x^{2} + \frac{8 x^{2}}{9} - 53 = 0$

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

To write $5 x^{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$5 x^{2} \cdot \frac{9}{9} + \frac{8 x^{2}}{9} - 53 = 0$

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

Simplify terms.

Combine $5 x^{2}$ and $\frac{9}{9}$ .

$\frac{5 x^{2} \cdot 9}{9} + \frac{8 x^{2}}{9} - 53 = 0$

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

Combine the numerators over the common denominator.

$\frac{5 x^{2} \cdot 9 + 8 x^{2}}{9} - 53 = 0$

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

$\frac{5 x^{2} \cdot 9 + 8 x^{2}}{9} - 53 = 0$

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

Simplify each term.

Simplify the numerator.

Factor $x^{2}$ out of $5 x^{2} \cdot 9 + 8 x^{2}$ .

Factor $x^{2}$ out of $5 x^{2} \cdot 9$ .

$\frac{x^{2} (5 \cdot 9) + 8 x^{2}}{9} - 53 = 0$

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

Factor $x^{2}$ out of $8 x^{2}$ .

$\frac{x^{2} (5 \cdot 9) + x^{2} \cdot 8}{9} - 53 = 0$

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

Factor $x^{2}$ out of $x^{2} (5 \cdot 9) + x^{2} \cdot 8$ .

$\frac{x^{2} (5 \cdot 9 + 8)}{9} - 53 = 0$

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

$\frac{x^{2} (5 \cdot 9 + 8)}{9} - 53 = 0$

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

Multiply $5$ by $9$ .

$\frac{x^{2} (45 + 8)}{9} - 53 = 0$

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

Add $45$ and $8$ .

$\frac{x^{2} \cdot 53}{9} - 53 = 0$

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

$\frac{x^{2} \cdot 53}{9} - 53 = 0$

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

Move $53$ to the left of $x^{2}$ .

$\frac{53 x^{2}}{9} - 53 = 0$

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

$\frac{53 x^{2}}{9} - 53 = 0$

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

$\frac{53 x^{2}}{9} - 53 = 0$

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

$\frac{53 x^{2}}{9} - 53 = 0$

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

$\frac{53 x^{2}}{9} - 53 = 0$

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

Step 3

Solve for $x$ in $\frac{53 x^{2}}{9} - 53 = 0$ .

Add $53$ to both sides of the equation.

$\frac{53 x^{2}}{9} = 53$

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

Multiply both sides of the equation by $\frac{9}{53}$ .

$\frac{9}{53} \cdot \frac{53 x^{2}}{9} = \frac{9}{53} \cdot 53$

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{9}{53} \cdot \frac{53 x^{2}}{9}$ .

Combine.

$\frac{9 (53 x^{2})}{53 \cdot 9} = \frac{9}{53} \cdot 53$

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

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 (53 x^{2})}{53 \cdot 9} = \frac{9}{53} \cdot 53$

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

Rewrite the expression.

$\frac{53 x^{2}}{53} = \frac{9}{53} \cdot 53$

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

$\frac{53 x^{2}}{53} = \frac{9}{53} \cdot 53$

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

Cancel the common factor of $53$ .

Cancel the common factor.

$\frac{53 x^{2}}{53} = \frac{9}{53} \cdot 53$

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

Divide $x^{2}$ by $1$ .