Solve {l}{2x+2y=42}{x^2+y^2=225}

$$\left. \begin{array} { l } { 2 x + 2 y = 42 } \\ { x ^ { 2 } + y ^ { 2 } = 225 } \end{array} \right.$$

Solve for x, y

$$x=12\text{, }y=9$$

$$x=9\text{, }y=12$$

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.

$$2x+2y=42,y^{2}+x^{2}=225$$

Solve 2x+2y=42 for x by isolating x on the left hand side of the equal sign.

$$2x+2y=42$$

Subtract 2y from both sides of the equation.

$$2x=-2y+42$$

Divide both sides by 2.

$$x=-y+21$$

Substitute -y+21 for x in the other equation, y^{2}+x^{2}=225.

$$y^{2}+\left(-y+21\right)^{2}=225$$

Square -y+21.

$$y^{2}+y^{2}-42y+441=225$$

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

$$2y^{2}-42y+441=225$$

Subtract 225 from both sides of the equation.

$$2y^{2}-42y+216=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\times 21\left(-1\right)\times 2 for b, and 216 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}-4\times 2\times 216}}{2\times 2}$$

Square 1\times 21\left(-1\right)\times 2.

$$y=\frac{-\left(-42\right)±\sqrt{1764-4\times 2\times 216}}{2\times 2}$$

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

$$y=\frac{-\left(-42\right)±\sqrt{1764-8\times 216}}{2\times 2}$$

Multiply -8 times 216.

$$y=\frac{-\left(-42\right)±\sqrt{1764-1728}}{2\times 2}$$

Add 1764 to -1728.

$$y=\frac{-\left(-42\right)±\sqrt{36}}{2\times 2}$$

Take the square root of 36.

$$y=\frac{-\left(-42\right)±6}{2\times 2}$$

The opposite of 1\times 21\left(-1\right)\times 2 is 42.

$$y=\frac{42±6}{2\times 2}$$

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

$$y=\frac{42±6}{4}$$

Now solve the equation y=\frac{42±6}{4} when ± is plus. Add 42 to 6.

$$y=\frac{48}{4}$$

Divide 48 by 4.

$$y=12$$

Now solve the equation y=\frac{42±6}{4} when ± is minus. Subtract 6 from 42.

$$y=\frac{36}{4}$$

Divide 36 by 4.

$$y=9$$

There are two solutions for y: 12 and 9. Substitute 12 for y in the equation x=-y+21 to find the corresponding solution for x that satisfies both equations.

$$x=-12+21$$

Add -12 to 21.

$$x=9$$

Now substitute 9 for y in the equation x=-y+21 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-9+21$$

Add -9 to 21.

$$x=12$$

The system is now solved.

$$x=9,y=12\text{ or }x=12,y=9$$