Solve {c}{a-3y/a-y=3/5}{2y=0.5}{xy/5=0.1}

$$\left. \begin{array} { c } { \frac { a - 3 y } { a - y } = \frac { 3 } { 5 } } \\ { 2 y = 0.5 } \\ { \frac { x y } { 5 } = 0.1 } \end{array} \right.$$

Solve for a, y, x

$$x=2$$

$$y=0.25$$

$$a=1.5$$

Short Solution Steps

Consider the second equation. Divide both sides by 2.

$$y=\frac{0.5}{2}$$

Expand \frac{0.5}{2} by multiplying both numerator and the denominator by 10.

$$y=\frac{5}{20}$$

Reduce the fraction \frac{5}{20} to lowest terms by extracting and canceling out 5.

$$y=\frac{1}{4}$$

Consider the third equation. Insert the known values of variables into the equation.

$$\frac{x\times \frac{1}{4}}{5}=0.1$$

Multiply both sides by 5.

$$x\times \frac{1}{4}=0.1\times 5$$

Multiply 0.1 and 5 to get 0.5.

$$x\times \frac{1}{4}=0.5$$

Multiply both sides by 4, the reciprocal of \frac{1}{4}.

$$x=0.5\times 4$$

Multiply 0.5 and 4 to get 2.

$$x=2$$

Consider the first equation. Insert the known values of variables into the equation.

$$\frac{a-3\times \frac{1}{4}}{a-\frac{1}{4}}=\frac{3}{5}$$

Multiply -3 and \frac{1}{4} to get -\frac{3}{4}.

$$\frac{a-\frac{3}{4}}{a-\frac{1}{4}}=\frac{3}{5}$$

Factor the expressions that are not already factored in \frac{a-\frac{3}{4}}{a-\frac{1}{4}}.

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

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{4a-3}{\left(\frac{1}{4}\right)^{0}\left(4a-1\right)}=\frac{3}{5}$$

Calculate \frac{1}{4} to the power of 0 and get 1.

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

Use the distributive property to multiply 1 by 4a-1.

$$\frac{4a-3}{4a-1}=\frac{3}{5}$$

Variable a cannot be equal to \frac{1}{4} since division by zero is not defined. Multiply both sides of the equation by 5\left(4a-1\right), the least common multiple of 4a-1,5.

$$5\left(4a-3\right)=3\left(4a-1\right)$$

Use the distributive property to multiply 5 by 4a-3.

$$20a-15=3\left(4a-1\right)$$

Use the distributive property to multiply 3 by 4a-1.

$$20a-15=12a-3$$

Subtract 12a from both sides.

$$20a-15-12a=-3$$

Combine 20a and -12a to get 8a.

$$8a-15=-3$$

Add 15 to both sides.

$$8a=-3+15$$

Add -3 and 15 to get 12.

$$8a=12$$

Divide both sides by 8.

$$a=\frac{12}{8}$$

Reduce the fraction \frac{12}{8} to lowest terms by extracting and canceling out 4.

$$a=\frac{3}{2}$$

The system is now solved.

$$a=\frac{3}{2} y=\frac{1}{4} x=2$$