Solve {l}{3[x-2(x-a/2)]=4x}{3x+a/12-1-5x/8=1}

$$\left. \begin{array} { l } { 3 [ x - 2 ( x - \frac { a } { 2 } ) ] = 4 x } \\ { \frac { 3 x + a } { 12 } - \frac { 1 - 5 x } { 8 } = 1 } \end{array} \right.$$

Solve for x, a

$$x = \frac{81}{77} = 1\frac{4}{77} \approx 1.051948052$$

$$a = \frac{27}{11} = 2\frac{5}{11} \approx 2.454545455$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 2.

$$6\left(x-2\left(x-\frac{a}{2}\right)\right)=8x$$

Subtract 8x from both sides.

$$6\left(x-2\left(x-\frac{a}{2}\right)\right)-8x=0$$

Multiply both sides of the equation by 2.

$$12\left(x-2\left(x-\frac{a}{2}\right)\right)-16x=0$$

Multiply both sides of the equation by 2.

$$24\left(x-2\left(x-\frac{a}{2}\right)\right)-32x=0$$

Use the distributive property to multiply -2 by x-\frac{a}{2}.

$$24\left(x-2x+2\times \frac{a}{2}\right)-32x=0$$

Express 2\times \frac{a}{2} as a single fraction.

$$24\left(x-2x+\frac{2a}{2}\right)-32x=0$$

Cancel out 2 and 2.

$$24\left(x-2x+a\right)-32x=0$$

Combine x and -2x to get -x.

$$24\left(-x+a\right)-32x=0$$

Use the distributive property to multiply 24 by -x+a.

$$-24x+24a-32x=0$$

Combine -24x and -32x to get -56x.

$$-56x+24a=0$$

Consider the second equation. Multiply both sides of the equation by 24, the least common multiple of 12,8.

$$2\left(3x+a\right)-3\left(1-5x\right)=24$$

Use the distributive property to multiply 2 by 3x+a.

$$6x+2a-3\left(1-5x\right)=24$$

Use the distributive property to multiply -3 by 1-5x.

$$6x+2a-3+15x=24$$

Combine 6x and 15x to get 21x.

$$21x+2a-3=24$$

Add 3 to both sides.

$$21x+2a=24+3$$

Add 24 and 3 to get 27.

$$21x+2a=27$$

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.

$$-56x+24a=0,21x+2a=27$$

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

$$-56x+24a=0$$

Subtract 24a from both sides of the equation.

$$-56x=-24a$$

Divide both sides by -56.

$$x=-\frac{1}{56}\left(-24\right)a$$

Multiply -\frac{1}{56} times -24a.

$$x=\frac{3}{7}a$$

Substitute \frac{3a}{7} for x in the other equation, 21x+2a=27.

$$21\times \frac{3}{7}a+2a=27$$

Multiply 21 times \frac{3a}{7}.

$$9a+2a=27$$

Add 9a to 2a.

$$11a=27$$

Divide both sides by 11.

$$a=\frac{27}{11}$$

Substitute \frac{27}{11} for a in x=\frac{3}{7}a. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{3}{7}\times \frac{27}{11}$$

Multiply \frac{3}{7} times \frac{27}{11} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{81}{77}$$

The system is now solved.

$$x=\frac{81}{77},a=\frac{27}{11}$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 2.

$$6\left(x-2\left(x-\frac{a}{2}\right)\right)=8x$$

Subtract 8x from both sides.

$$6\left(x-2\left(x-\frac{a}{2}\right)\right)-8x=0$$

Multiply both sides of the equation by 2.

$$12\left(x-2\left(x-\frac{a}{2}\right)\right)-16x=0$$

Multiply both sides of the equation by 2.

$$24\left(x-2\left(x-\frac{a}{2}\right)\right)-32x=0$$

Use the distributive property to multiply -2 by x-\frac{a}{2}.

$$24\left(x-2x+2\times \frac{a}{2}\right)-32x=0$$

Express 2\times \frac{a}{2} as a single fraction.

$$24\left(x-2x+\frac{2a}{2}\right)-32x=0$$

Cancel out 2 and 2.

$$24\left(x-2x+a\right)-32x=0$$

Combine x and -2x to get -x.

$$24\left(-x+a\right)-32x=0$$

Use the distributive property to multiply 24 by -x+a.

$$-24x+24a-32x=0$$

Combine -24x and -32x to get -56x.

$$-56x+24a=0$$

Consider the second equation. Multiply both sides of the equation by 24, the least common multiple of 12,8.

$$2\left(3x+a\right)-3\left(1-5x\right)=24$$

Use the distributive property to multiply 2 by 3x+a.

$$6x+2a-3\left(1-5x\right)=24$$

Use the distributive property to multiply -3 by 1-5x.

$$6x+2a-3+15x=24$$

Combine 6x and 15x to get 21x.

$$21x+2a-3=24$$

Add 3 to both sides.

$$21x+2a=24+3$$

Add 24 and 3 to get 27.

$$21x+2a=27$$

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

$$-56x+24a=0,21x+2a=27$$

Write the equations in matrix form.

$$\left(\begin{matrix}-56&24\\21&2\end{matrix}\right)\left(\begin{matrix}x\\a\end{matrix}\right)=\left(\begin{matrix}0\\27\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-56&24\\21&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-56&24\\21&2\end{matrix}\right))\left(\begin{matrix}-56&24\\21&2\end{matrix}\right)\left(\begin{matrix}x\\a\end{matrix}\right)=inverse(\left(\begin{matrix}-56&24\\21&2\end{matrix}\right))\left(\begin{matrix}0\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\a\end{matrix}\right)=inverse(\left(\begin{matrix}-56&24\\21&2\end{matrix}\right))\left(\begin{matrix}0\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\a\end{matrix}\right)=inverse(\left(\begin{matrix}-56&24\\21&2\end{matrix}\right))\left(\begin{matrix}0\\27\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\a\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-56\times 2-24\times 21}&-\frac{24}{-56\times 2-24\times 21}\\-\frac{21}{-56\times 2-24\times 21}&-\frac{56}{-56\times 2-24\times 21}\end{matrix}\right)\left(\begin{matrix}0\\27\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\a\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{308}&\frac{3}{77}\\\frac{3}{88}&\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}0\\27\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\a\end{matrix}\right)=\left(\begin{matrix}\frac{3}{77}\times 27\\\frac{1}{11}\times 27\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\a\end{matrix}\right)=\left(\begin{matrix}\frac{81}{77}\\\frac{27}{11}\end{matrix}\right)$$

Extract the matrix elements x and a.

$$x=\frac{81}{77},a=\frac{27}{11}$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 2.

$$6\left(x-2\left(x-\frac{a}{2}\right)\right)=8x$$

Subtract 8x from both sides.

$$6\left(x-2\left(x-\frac{a}{2}\right)\right)-8x=0$$

Multiply both sides of the equation by 2.

$$12\left(x-2\left(x-\frac{a}{2}\right)\right)-16x=0$$

Multiply both sides of the equation by 2.

$$24\left(x-2\left(x-\frac{a}{2}\right)\right)-32x=0$$

Use the distributive property to multiply -2 by x-\frac{a}{2}.

$$24\left(x-2x+2\times \frac{a}{2}\right)-32x=0$$

Express 2\times \frac{a}{2} as a single fraction.

$$24\left(x-2x+\frac{2a}{2}\right)-32x=0$$

Cancel out 2 and 2.

$$24\left(x-2x+a\right)-32x=0$$

Combine x and -2x to get -x.

$$24\left(-x+a\right)-32x=0$$

Use the distributive property to multiply 24 by -x+a.

$$-24x+24a-32x=0$$

Combine -24x and -32x to get -56x.

$$-56x+24a=0$$

Consider the second equation. Multiply both sides of the equation by 24, the least common multiple of 12,8.

$$2\left(3x+a\right)-3\left(1-5x\right)=24$$

Use the distributive property to multiply 2 by 3x+a.

$$6x+2a-3\left(1-5x\right)=24$$

Use the distributive property to multiply -3 by 1-5x.

$$6x+2a-3+15x=24$$

Combine 6x and 15x to get 21x.

$$21x+2a-3=24$$

Add 3 to both sides.

$$21x+2a=24+3$$

Add 24 and 3 to get 27.

$$21x+2a=27$$

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.

$$-56x+24a=0,21x+2a=27$$

To make -56x and 21x equal, multiply all terms on each side of the first equation by 21 and all terms on each side of the second by -56.

$$21\left(-56\right)x+21\times 24a=0,-56\times 21x-56\times 2a=-56\times 27$$

Simplify.

$$-1176x+504a=0,-1176x-112a=-1512$$

Subtract -1176x-112a=-1512 from -1176x+504a=0 by subtracting like terms on each side of the equal sign.

$$-1176x+1176x+504a+112a=1512$$

Add -1176x to 1176x. Terms -1176x and 1176x cancel out, leaving an equation with only one variable that can be solved.

$$504a+112a=1512$$

Add 504a to 112a.

$$616a=1512$$

Divide both sides by 616.

$$a=\frac{27}{11}$$

Substitute \frac{27}{11} for a in 21x+2a=27. Because the resulting equation contains only one variable, you can solve for x directly.

$$21x+2\times \frac{27}{11}=27$$

Multiply 2 times \frac{27}{11}.

$$21x+\frac{54}{11}=27$$

Subtract \frac{54}{11} from both sides of the equation.

$$21x=\frac{243}{11}$$

Divide both sides by 21.

$$x=\frac{81}{77}$$

The system is now solved.

$$x=\frac{81}{77},a=\frac{27}{11}$$