Solve {l}{3x+4y+m=0}{9x^2+16y^2=144}

$$\left. \begin{array} { l } { 3 x + 4 y + m = 0 } \\ { 9 x ^ { 2 } + 16 y ^ { 2 } = 144 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{\sqrt{288-m^{2}}-m}{6}\text{, }y=\frac{-\sqrt{288-m^{2}}-m}{8}$$

$$x=\frac{-\sqrt{288-m^{2}}-m}{6}\text{, }y=\frac{\sqrt{288-m^{2}}-m}{8}$$

Steps Using Substitution and Quadratic Formula

Solve 3x+4y+m=0 for x by isolating x on the left hand side of the equal sign.

$$3x+4y+m=0$$

Subtract m from both sides of the equation.

$$3x+4y=-m$$

Subtract 4y from both sides of the equation.

$$3x=-4y-m$$

Divide both sides by 3.

$$x=-\frac{4}{3}y-\frac{m}{3}$$

Substitute -\frac{4}{3}y-\frac{m}{3} for x in the other equation, 16y^{2}+9x^{2}=144.

$$16y^{2}+9\left(-\frac{4}{3}y-\frac{m}{3}\right)^{2}=144$$

Square -\frac{4}{3}y-\frac{m}{3}.

$$16y^{2}+9\left(\frac{16}{9}y^{2}+\left(-\frac{8}{3}\left(-\frac{m}{3}\right)\right)y+\left(-\frac{m}{3}\right)^{2}\right)=144$$

Multiply 9 times \frac{16}{9}y^{2}+\left(-\frac{8}{3}\left(-\frac{m}{3}\right)\right)y+\left(-\frac{m}{3}\right)^{2}.

$$16y^{2}+16y^{2}+\left(-24\left(-\frac{m}{3}\right)\right)y+9\left(-\frac{m}{3}\right)^{2}=144$$

Add 16y^{2} to 16y^{2}.

$$32y^{2}+\left(-24\left(-\frac{m}{3}\right)\right)y+9\left(-\frac{m}{3}\right)^{2}=144$$

Subtract 144 from both sides of the equation.

$$32y^{2}+\left(-24\left(-\frac{m}{3}\right)\right)y+9\left(-\frac{m}{3}\right)^{2}-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16+9\left(-\frac{4}{3}\right)^{2} for a, 9\left(-\frac{4}{3}\right)\times 2\left(-\frac{m}{3}\right) for b, and -144+m^{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{\left(-24\left(-\frac{m}{3}\right)\right)^{2}-4\times 32\left(m^{2}-144\right)}}{2\times 32}$$

Square 9\left(-\frac{4}{3}\right)\times 2\left(-\frac{m}{3}\right).

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{64m^{2}-4\times 32\left(m^{2}-144\right)}}{2\times 32}$$

Multiply -4 times 16+9\left(-\frac{4}{3}\right)^{2}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{64m^{2}-128\left(m^{2}-144\right)}}{2\times 32}$$

Multiply -128 times -144+m^{2}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{64m^{2}[phone]m^{2}}}{2\times 32}$$

Add 64m^{2} to 18432-128m^{2}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{18432-64m^{2}}}{2\times 32}$$

Take the square root of 18432-64m^{2}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±8\sqrt{288-m^{2}}}{2\times 32}$$

Multiply 2 times 16+9\left(-\frac{4}{3}\right)^{2}.

$$y=\frac{-8m±8\sqrt{288-m^{2}}}{64}$$

Now solve the equation y=\frac{-8m±8\sqrt{288-m^{2}}}{64} when ± is plus. Add -8m to 8\sqrt{288-m^{2}}.

$$y=\frac{8\sqrt{288-m^{2}}-8m}{64}$$

Divide -8m+8\sqrt{288-m^{2}} by 64.

$$y=\frac{\sqrt{288-m^{2}}-m}{8}$$

Now solve the equation y=\frac{-8m±8\sqrt{288-m^{2}}}{64} when ± is minus. Subtract 8\sqrt{288-m^{2}} from -8m.

$$y=\frac{-8\sqrt{288-m^{2}}-8m}{64}$$

Divide -8m-8\sqrt{288-m^{2}} by 64.

$$y=\frac{-\sqrt{288-m^{2}}-m}{8}$$

There are two solutions for y: \frac{-m+\sqrt{288-m^{2}}}{8} and \frac{-m-\sqrt{288-m^{2}}}{8}. Substitute \frac{-m+\sqrt{288-m^{2}}}{8} for y in the equation x=-\frac{4}{3}y-\frac{m}{3} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\times \frac{\sqrt{288-m^{2}}-m}{8}-\frac{m}{3}$$

Multiply -\frac{4}{3} times \frac{-m+\sqrt{288-m^{2}}}{8}.

$$x=\frac{-4\times \frac{\sqrt{288-m^{2}}-m}{8}-m}{3}$$

Now substitute \frac{-m-\sqrt{288-m^{2}}}{8} for y in the equation x=-\frac{4}{3}y-\frac{m}{3} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\times \frac{-\sqrt{288-m^{2}}-m}{8}-\frac{m}{3}$$

Multiply -\frac{4}{3} times \frac{-m-\sqrt{288-m^{2}}}{8}.

$$x=\frac{-4\times \frac{-\sqrt{288-m^{2}}-m}{8}-m}{3}$$

The system is now solved.

$$x=\frac{-4\times \frac{\sqrt{288-m^{2}}-m}{8}-m}{3},y=\frac{\sqrt{288-m^{2}}-m}{8}\text{ or }x=\frac{-4\times \frac{-\sqrt{288-m^{2}}-m}{8}-m}{3},y=\frac{-\sqrt{288-m^{2}}-m}{8}$$

Solve for x, y

$$x=\frac{\sqrt{288-m^{2}}-m}{6}\text{, }y=\frac{-\sqrt{288-m^{2}}-m}{8}$$

$$x=\frac{-\sqrt{288-m^{2}}-m}{6}\text{, }y=\frac{\sqrt{288-m^{2}}-m}{8}\text{, }|m|\leq 12\sqrt{2}$$

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.

$$3x+4y+m=0,16y^{2}+9x^{2}=144$$

Solve 3x+4y+m=0 for x by isolating x on the left hand side of the equal sign.

$$3x+4y+m=0$$

Subtract m from both sides of the equation.

$$3x+4y=-m$$

Subtract 4y from both sides of the equation.

$$3x=-4y-m$$

Divide both sides by 3.

$$x=-\frac{4}{3}y-\frac{m}{3}$$

Substitute -\frac{4}{3}y-\frac{m}{3} for x in the other equation, 16y^{2}+9x^{2}=144.

$$16y^{2}+9\left(-\frac{4}{3}y-\frac{m}{3}\right)^{2}=144$$

Square -\frac{4}{3}y-\frac{m}{3}.

$$16y^{2}+9\left(\frac{16}{9}y^{2}+\left(-\frac{8}{3}\left(-\frac{m}{3}\right)\right)y+\left(-\frac{m}{3}\right)^{2}\right)=144$$

Multiply 9 times \frac{16}{9}y^{2}+\left(-\frac{8}{3}\left(-\frac{m}{3}\right)\right)y+\left(-\frac{m}{3}\right)^{2}.

$$16y^{2}+16y^{2}+\left(-24\left(-\frac{m}{3}\right)\right)y+9\left(-\frac{m}{3}\right)^{2}=144$$

Add 16y^{2} to 16y^{2}.

$$32y^{2}+\left(-24\left(-\frac{m}{3}\right)\right)y+9\left(-\frac{m}{3}\right)^{2}=144$$

Subtract 144 from both sides of the equation.

$$32y^{2}+\left(-24\left(-\frac{m}{3}\right)\right)y+9\left(-\frac{m}{3}\right)^{2}-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16+9\left(-\frac{4}{3}\right)^{2} for a, 9\left(-\frac{4}{3}\right)\times 2\left(-\frac{m}{3}\right) for b, and -144+m^{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{\left(-24\left(-\frac{m}{3}\right)\right)^{2}-4\times 32\left(m^{2}-144\right)}}{2\times 32}$$

Square 9\left(-\frac{4}{3}\right)\times 2\left(-\frac{m}{3}\right).

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{64m^{2}-4\times 32\left(m^{2}-144\right)}}{2\times 32}$$

Multiply -4 times 16+9\left(-\frac{4}{3}\right)^{2}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{64m^{2}-128\left(m^{2}-144\right)}}{2\times 32}$$

Multiply -128 times -144+m^{2}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{64m^{2}[phone]m^{2}}}{2\times 32}$$

Add 64m^{2} to 18432-128m^{2}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±\sqrt{18432-64m^{2}}}{2\times 32}$$

Take the square root of 18432-64m^{2}.

$$y=\frac{-\left(-24\left(-\frac{m}{3}\right)\right)±8\sqrt{288-m^{2}}}{2\times 32}$$

Multiply 2 times 16+9\left(-\frac{4}{3}\right)^{2}.

$$y=\frac{-8m±8\sqrt{288-m^{2}}}{64}$$

Now solve the equation y=\frac{-8m±8\sqrt{288-m^{2}}}{64} when ± is plus. Add -8m to 8\sqrt{288-m^{2}}.

$$y=\frac{8\sqrt{288-m^{2}}-8m}{64}$$

Divide -8m+8\sqrt{288-m^{2}} by 64.

$$y=\frac{\sqrt{288-m^{2}}-m}{8}$$

Now solve the equation y=\frac{-8m±8\sqrt{288-m^{2}}}{64} when ± is minus. Subtract 8\sqrt{288-m^{2}} from -8m.

$$y=\frac{-8\sqrt{288-m^{2}}-8m}{64}$$

Divide -8m-8\sqrt{288-m^{2}} by 64.

$$y=\frac{-\sqrt{288-m^{2}}-m}{8}$$

There are two solutions for y: \frac{-m+\sqrt{288-m^{2}}}{8} and \frac{-m-\sqrt{288-m^{2}}}{8}. Substitute \frac{-m+\sqrt{288-m^{2}}}{8} for y in the equation x=-\frac{4}{3}y-\frac{m}{3} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\times \frac{\sqrt{288-m^{2}}-m}{8}-\frac{m}{3}$$

Multiply -\frac{4}{3} times \frac{-m+\sqrt{288-m^{2}}}{8}.

$$x=\frac{-4\times \frac{\sqrt{288-m^{2}}-m}{8}-m}{3}$$

Now substitute \frac{-m-\sqrt{288-m^{2}}}{8} for y in the equation x=-\frac{4}{3}y-\frac{m}{3} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\times \frac{-\sqrt{288-m^{2}}-m}{8}-\frac{m}{3}$$

Multiply -\frac{4}{3} times \frac{-m-\sqrt{288-m^{2}}}{8}.

$$x=\frac{-4\times \frac{-\sqrt{288-m^{2}}-m}{8}-m}{3}$$

The system is now solved.

$$x=\frac{-4\times \frac{\sqrt{288-m^{2}}-m}{8}-m}{3},y=\frac{\sqrt{288-m^{2}}-m}{8}\text{ or }x=\frac{-4\times \frac{-\sqrt{288-m^{2}}-m}{8}-m}{3},y=\frac{-\sqrt{288-m^{2}}-m}{8}$$