Solve {l}{x+5/2-3x-4/3=x/2+23/6}{frac{2}{3}(a-3)+frac{1}{2}(a+1)=frac{5}{2}(a+2)-frac{1}{4}}

$$\left. \begin{array} { l } { \frac { x + 5 } { 2 } - \frac { 3 x - 4 } { 3 } = \frac { x } { 2 } + \frac { 23 } { 6 } } \\ { \frac { 2 } { 3 } ( a - 3 ) + \frac { 1 } { 2 } ( a + 1 ) = \frac { 5 } { 2 } ( a + 2 ) - \frac { 1 } { 4 } } \end{array} \right.$$

Solve for x, a

$$x=0$$

$$a = -\frac{75}{16} = -4\frac{11}{16} = -4.6875$$

Solution Steps

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3,6.

$$3\left(x+5\right)-2\left(3x-4\right)=3x+23$$

Use the distributive property to multiply 3 by x+5.

$$3x+15-2\left(3x-4\right)=3x+23$$

Use the distributive property to multiply -2 by 3x-4.

$$3x+15-6x+8=3x+23$$

Combine 3x and -6x to get -3x.

$$-3x+15+8=3x+23$$

Add 15 and 8 to get 23.

$$-3x+23=3x+23$$

Subtract 3x from both sides.

$$-3x+23-3x=23$$

Combine -3x and -3x to get -6x.

$$-6x+23=23$$

Subtract 23 from both sides.

$$-6x=23-23$$

Subtract 23 from 23 to get 0.

$$-6x=0$$

Divide both sides by -6. Zero divided by any non-zero number gives zero.

$$x=0$$

Consider the second equation. Use the distributive property to multiply \frac{2}{3} by a-3.

$$\frac{2}{3}a-2+\frac{1}{2}\left(a+1\right)=\frac{5}{2}\left(a+2\right)-\frac{1}{4}$$

Use the distributive property to multiply \frac{1}{2} by a+1.

$$\frac{2}{3}a-2+\frac{1}{2}a+\frac{1}{2}=\frac{5}{2}\left(a+2\right)-\frac{1}{4}$$

Combine \frac{2}{3}a and \frac{1}{2}a to get \frac{7}{6}a.

$$\frac{7}{6}a-2+\frac{1}{2}=\frac{5}{2}\left(a+2\right)-\frac{1}{4}$$

Add -2 and \frac{1}{2} to get -\frac{3}{2}.

$$\frac{7}{6}a-\frac{3}{2}=\frac{5}{2}\left(a+2\right)-\frac{1}{4}$$

Use the distributive property to multiply \frac{5}{2} by a+2.

$$\frac{7}{6}a-\frac{3}{2}=\frac{5}{2}a+5-\frac{1}{4}$$

Subtract \frac{1}{4} from 5 to get \frac{19}{4}.

$$\frac{7}{6}a-\frac{3}{2}=\frac{5}{2}a+\frac{19}{4}$$

Subtract \frac{5}{2}a from both sides.

$$\frac{7}{6}a-\frac{3}{2}-\frac{5}{2}a=\frac{19}{4}$$

Combine \frac{7}{6}a and -\frac{5}{2}a to get -\frac{4}{3}a.

$$-\frac{4}{3}a-\frac{3}{2}=\frac{19}{4}$$

Add \frac{3}{2} to both sides.

$$-\frac{4}{3}a=\frac{19}{4}+\frac{3}{2}$$

Add \frac{19}{4} and \frac{3}{2} to get \frac{25}{4}.

$$-\frac{4}{3}a=\frac{25}{4}$$

Multiply both sides by -\frac{3}{4}, the reciprocal of -\frac{4}{3}.

$$a=\frac{25}{4}\left(-\frac{3}{4}\right)$$

Multiply \frac{25}{4} and -\frac{3}{4} to get -\frac{75}{16}.

$$a=-\frac{75}{16}$$

The system is now solved.

$$x=0 a=-\frac{75}{16}$$