Solve {l}{4x-2y-6=0}{4(8+x)+2(20y)-26=0}

$$\left\{ \begin{array} { l } { 4 x - 2 y - 6 = 0 } \\ { 4 ( 8 + x ) + 2 ( 20 y ) - 26 = 0 } \end{array} \right.$$

Solve for x, y

$$x = \frac{19}{14} = 1\frac{5}{14} \approx 1.357142857$$

$$y=-\frac{2}{7}\approx -0.285714286$$

Steps Using Substitution

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.

$$4x-2y-6=0,4\left(x+8\right)+40y-26=0$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$4x-2y-6=0$$

Add 6 to both sides of the equation.

$$4x-2y=6$$

Add 2y to both sides of the equation.

$$4x=2y+6$$

Divide both sides by 4.

$$x=\frac{1}{4}\left(2y+6\right)$$

Multiply \frac{1}{4} times 6+2y.

$$x=\frac{1}{2}y+\frac{3}{2}$$

Substitute \frac{3+y}{2} for x in the other equation, 4\left(x+8\right)+40y-26=0.

$$4\left(\frac{1}{2}y+\frac{3}{2}+8\right)+40y-26=0$$

Add \frac{3}{2} to 8.

$$4\left(\frac{1}{2}y+\frac{19}{2}\right)+40y-26=0$$

Multiply 4 times \frac{19+y}{2}.

$$2y+38+40y-26=0$$

Add 2y to 40y.

$$42y+38-26=0$$

Add 38 to -26.

$$42y+12=0$$

Subtract 12 from both sides of the equation.

$$42y=-12$$

Divide both sides by 42.

$$y=-\frac{2}{7}$$

Substitute -\frac{2}{7} for y in x=\frac{1}{2}y+\frac{3}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{1}{2}\left(-\frac{2}{7}\right)+\frac{3}{2}$$

Multiply \frac{1}{2} times -\frac{2}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{1}{7}+\frac{3}{2}$$

Add \frac{3}{2} to -\frac{1}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{19}{14}$$

The system is now solved.

$$x=\frac{19}{14},y=-\frac{2}{7}$$

Steps Using Matrices

Put the equations in standard form and then use matrices to solve the system of equations.

$$4x-2y-6=0,4\left(x+8\right)+40y-26=0$$

Simplify the second equation to put it in standard form.

$$4\left(x+8\right)+40y-26=0$$

Multiply 4 times x+8.

$$4x+32+40y-26=0$$

Add 32 to -26.

$$4x+40y+6=0$$

Subtract 6 from both sides of the equation.

$$4x+40y=-6$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&-2\\4&40\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&-2\\4&40\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-2\\4&40\end{matrix}\right))\left(\begin{matrix}4&-2\\4&40\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\4&40\end{matrix}\right))\left(\begin{matrix}6\\-6\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\4&40\end{matrix}\right))\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\4&40\end{matrix}\right))\left(\begin{matrix}6\\-6\end{matrix}\right)$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{40}{4\times 40-\left(-2\times 4\right)}&-\frac{-2}{4\times 40-\left(-2\times 4\right)}\\-\frac{4}{4\times 40-\left(-2\times 4\right)}&\frac{4}{4\times 40-\left(-2\times 4\right)}\end{matrix}\right)\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}&\frac{1}{84}\\-\frac{1}{42}&\frac{1}{42}\end{matrix}\right)\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}\times 6+\frac{1}{84}\left(-6\right)\\-\frac{1}{42}\times 6+\frac{1}{42}\left(-6\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{14}\\-\frac{2}{7}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{19}{14},y=-\frac{2}{7}$$