Solve {l}{a=15-2b}{2a-5b=15-2a}

$$\left. \begin{array} { l } { a = 15 - 2 b } \\ { 2 a - 5 b = 15 - 2 a } \end{array} \right.$$

Solve for a, b

$$a = \frac{105}{13} = 8\frac{1}{13} \approx 8.076923077$$

$$b = \frac{45}{13} = 3\frac{6}{13} \approx 3.461538462$$

Steps Using Substitution

Consider the first equation. Add 2b to both sides.

$$a+2b=15$$

Consider the second equation. Add 2a to both sides.

$$2a-5b+2a=15$$

Combine 2a and 2a to get 4a.

$$4a-5b=15$$

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.

$$a+2b=15,4a-5b=15$$

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

$$a+2b=15$$

Subtract 2b from both sides of the equation.

$$a=-2b+15$$

Substitute -2b+15 for a in the other equation, 4a-5b=15.

$$4\left(-2b+15\right)-5b=15$$

Multiply 4 times -2b+15.

$$-8b+60-5b=15$$

Add -8b to -5b.

$$-13b+60=15$$

Subtract 60 from both sides of the equation.

$$-13b=-45$$

Divide both sides by -13.

$$b=\frac{45}{13}$$

Substitute \frac{45}{13} for b in a=-2b+15. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=-2\times \frac{45}{13}+15$$

Multiply -2 times \frac{45}{13}.

$$a=-\frac{90}{13}+15$$

Add 15 to -\frac{90}{13}.

$$a=\frac{105}{13}$$

The system is now solved.

$$a=\frac{105}{13},b=\frac{45}{13}$$

Steps Using Matrices

Consider the first equation. Add 2b to both sides.

$$a+2b=15$$

Consider the second equation. Add 2a to both sides.

$$2a-5b+2a=15$$

Combine 2a and 2a to get 4a.

$$4a-5b=15$$

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

$$a+2b=15,4a-5b=15$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&2\\4&-5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}15\\15\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&2\\4&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\4&-5\end{matrix}\right))\left(\begin{matrix}1&2\\4&-5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&-5\end{matrix}\right))\left(\begin{matrix}15\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&-5\end{matrix}\right))\left(\begin{matrix}15\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&-5\end{matrix}\right))\left(\begin{matrix}15\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-5-2\times 4}&-\frac{2}{-5-2\times 4}\\-\frac{4}{-5-2\times 4}&\frac{1}{-5-2\times 4}\end{matrix}\right)\left(\begin{matrix}15\\15\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{5}{13}&\frac{2}{13}\\\frac{4}{13}&-\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}15\\15\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{5}{13}\times 15+\frac{2}{13}\times 15\\\frac{4}{13}\times 15-\frac{1}{13}\times 15\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{105}{13}\\\frac{45}{13}\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=\frac{105}{13},b=\frac{45}{13}$$

Steps Using Elimination

Consider the first equation. Add 2b to both sides.

$$a+2b=15$$

Consider the second equation. Add 2a to both sides.

$$2a-5b+2a=15$$

Combine 2a and 2a to get 4a.

$$4a-5b=15$$

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$a+2b=15,4a-5b=15$$

To make a and 4a equal, multiply all terms on each side of the first equation by 4 and all terms on each side of the second by 1.

$$4a+4\times 2b=4\times 15,4a-5b=15$$

Simplify.

$$4a+8b=60,4a-5b=15$$

Subtract 4a-5b=15 from 4a+8b=60 by subtracting like terms on each side of the equal sign.

$$4a-4a+8b+5b=60-15$$

Add 4a to -4a. Terms 4a and -4a cancel out, leaving an equation with only one variable that can be solved.

$$8b+5b=60-15$$

Add 8b to 5b.

$$13b=60-15$$

Add 60 to -15.

$$13b=45$$

Divide both sides by 13.

$$b=\frac{45}{13}$$

Substitute \frac{45}{13} for b in 4a-5b=15. Because the resulting equation contains only one variable, you can solve for a directly.

$$4a-5\times \frac{45}{13}=15$$

Multiply -5 times \frac{45}{13}.

$$4a-\frac{225}{13}=15$$

Add \frac{225}{13} to both sides of the equation.

$$4a=\frac{420}{13}$$

Divide both sides by 4.

$$a=\frac{105}{13}$$

The system is now solved.

$$a=\frac{105}{13},b=\frac{45}{13}$$