Solve {l}{0.3(0.5x+2)+0.4(0.2y-1)=-0.09}{4x-y/5-x-2y+1/4=-13/10-a}

$$\left\{ \begin{array} { l } { 0.3 ( 0.5 x + 2 ) + 0.4 ( 0.2 y - 1 ) = - 0.09 } \\ { \frac { 4 x - y } { 5 } - \frac { x - 2 y + 1 } { 4 } = - \frac { 13 } { 10 } - a } \end{array} \right.$$

Solve for x, y

$$x=80a-3$$

$$y=2-150a$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 0.3 by 0.5x+2.

$$0.15x+0.6+0.4\left(0.2y-1\right)=-0.09$$

Use the distributive property to multiply 0.4 by 0.2y-1.

$$0.15x+0.6+0.08y-0.4=-0.09$$

Subtract 0.4 from 0.6 to get 0.2.

$$0.15x+0.2+0.08y=-0.09$$

Subtract 0.2 from both sides.

$$0.15x+0.08y=-0.09-0.2$$

Subtract 0.2 from -0.09 to get -0.29.

$$0.15x+0.08y=-0.29$$

Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,4,10.

$$4\left(4x-y\right)-5\left(x-2y+1\right)=-26-20a$$

Use the distributive property to multiply 4 by 4x-y.

$$16x-4y-5\left(x-2y+1\right)=-26-20a$$

Use the distributive property to multiply -5 by x-2y+1.

$$16x-4y-5x+10y-5=-26-20a$$

Combine 16x and -5x to get 11x.

$$11x-4y+10y-5=-26-20a$$

Combine -4y and 10y to get 6y.

$$11x+6y-5=-26-20a$$

Add 5 to both sides.

$$11x+6y=-26-20a+5$$

Add -26 and 5 to get -21.

$$11x+6y=-21-20a$$

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.

$$0.15x+0.08y=-0.29,11x+6y=-20a-21$$

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

$$0.15x+0.08y=-0.29$$

Subtract \frac{2y}{25} from both sides of the equation.

$$0.15x=-0.08y-0.29$$

Divide both sides of the equation by 0.15, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{20}{3}\left(-0.08y-0.29\right)$$

Multiply \frac{20}{3} times -\frac{2y}{25}-0.29.

$$x=-\frac{8}{15}y-\frac{29}{15}$$

Substitute \frac{-8y-29}{15} for x in the other equation, 11x+6y=-20a-21.

$$11\left(-\frac{8}{15}y-\frac{29}{15}\right)+6y=-20a-21$$

Multiply 11 times \frac{-8y-29}{15}.

$$-\frac{88}{15}y-\frac{319}{15}+6y=-20a-21$$

Add -\frac{88y}{15} to 6y.

$$\frac{2}{15}y-\frac{319}{15}=-20a-21$$

Add \frac{319}{15} to both sides of the equation.

$$\frac{2}{15}y=\frac{4}{15}-20a$$

Divide both sides of the equation by \frac{2}{15}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2-150a$$

Substitute 2-150a for y in x=-\frac{8}{15}y-\frac{29}{15}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{8}{15}\left(2-150a\right)-\frac{29}{15}$$

Multiply -\frac{8}{15} times 2-150a.

$$x=80a-\frac{16}{15}-\frac{29}{15}$$

Add -\frac{29}{15} to -\frac{16}{15}+80a.

$$x=80a-3$$

The system is now solved.

$$x=80a-3,y=2-150a$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 0.3 by 0.5x+2.

$$0.15x+0.6+0.4\left(0.2y-1\right)=-0.09$$

Use the distributive property to multiply 0.4 by 0.2y-1.

$$0.15x+0.6+0.08y-0.4=-0.09$$

Subtract 0.4 from 0.6 to get 0.2.

$$0.15x+0.2+0.08y=-0.09$$

Subtract 0.2 from both sides.

$$0.15x+0.08y=-0.09-0.2$$

Subtract 0.2 from -0.09 to get -0.29.

$$0.15x+0.08y=-0.29$$

Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,4,10.

$$4\left(4x-y\right)-5\left(x-2y+1\right)=-26-20a$$

Use the distributive property to multiply 4 by 4x-y.

$$16x-4y-5\left(x-2y+1\right)=-26-20a$$

Use the distributive property to multiply -5 by x-2y+1.

$$16x-4y-5x+10y-5=-26-20a$$

Combine 16x and -5x to get 11x.

$$11x-4y+10y-5=-26-20a$$

Combine -4y and 10y to get 6y.

$$11x+6y-5=-26-20a$$

Add 5 to both sides.

$$11x+6y=-26-20a+5$$

Add -26 and 5 to get -21.

$$11x+6y=-21-20a$$

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

$$0.15x+0.08y=-0.29,11x+6y=-20a-21$$

Write the equations in matrix form.

$$\left(\begin{matrix}0.15&0.08\\11&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.29\\-20a-21\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}0.15&0.08\\11&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.15&0.08\\11&6\end{matrix}\right))\left(\begin{matrix}0.15&0.08\\11&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.15&0.08\\11&6\end{matrix}\right))\left(\begin{matrix}-0.29\\-20a-21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.15&0.08\\11&6\end{matrix}\right))\left(\begin{matrix}-0.29\\-20a-21\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.15&0.08\\11&6\end{matrix}\right))\left(\begin{matrix}-0.29\\-20a-21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{0.15\times 6-0.08\times 11}&-\frac{0.08}{0.15\times 6-0.08\times 11}\\-\frac{11}{0.15\times 6-0.08\times 11}&\frac{0.15}{0.15\times 6-0.08\times 11}\end{matrix}\right)\left(\begin{matrix}-0.29\\-20a-21\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}300&-4\\-550&7.5\end{matrix}\right)\left(\begin{matrix}-0.29\\-20a-21\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}300\left(-0.29\right)-4\left(-20a-21\right)\\-550\left(-0.29\right)+7.5\left(-20a-21\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}80a-3\\2-150a\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=80a-3,y=2-150a$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 0.3 by 0.5x+2.

$$0.15x+0.6+0.4\left(0.2y-1\right)=-0.09$$

Use the distributive property to multiply 0.4 by 0.2y-1.

$$0.15x+0.6+0.08y-0.4=-0.09$$

Subtract 0.4 from 0.6 to get 0.2.

$$0.15x+0.2+0.08y=-0.09$$

Subtract 0.2 from both sides.

$$0.15x+0.08y=-0.09-0.2$$

Subtract 0.2 from -0.09 to get -0.29.

$$0.15x+0.08y=-0.29$$

Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,4,10.

$$4\left(4x-y\right)-5\left(x-2y+1\right)=-26-20a$$

Use the distributive property to multiply 4 by 4x-y.

$$16x-4y-5\left(x-2y+1\right)=-26-20a$$

Use the distributive property to multiply -5 by x-2y+1.

$$16x-4y-5x+10y-5=-26-20a$$

Combine 16x and -5x to get 11x.

$$11x-4y+10y-5=-26-20a$$

Combine -4y and 10y to get 6y.

$$11x+6y-5=-26-20a$$

Add 5 to both sides.

$$11x+6y=-26-20a+5$$

Add -26 and 5 to get -21.

$$11x+6y=-21-20a$$

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.

$$0.15x+0.08y=-0.29,11x+6y=-20a-21$$

To make \frac{3x}{20} and 11x equal, multiply all terms on each side of the first equation by 11 and all terms on each side of the second by 0.15.

$$11\times 0.15x+11\times 0.08y=11\left(-0.29\right),0.15\times 11x+0.15\times 6y=0.15\left(-20a-21\right)$$

Simplify.

$$1.65x+0.88y=-3.19,1.65x+0.9y=-3a-3.15$$

Subtract 1.65x+0.9y=-3a-3.15 from 1.65x+0.88y=-3.19 by subtracting like terms on each side of the equal sign.

$$1.65x-1.65x+0.88y-0.9y=-3.19+3a+3.15$$

Add \frac{33x}{20} to -\frac{33x}{20}. Terms \frac{33x}{20} and -\frac{33x}{20} cancel out, leaving an equation with only one variable that can be solved.

$$0.88y-0.9y=-3.19+3a+3.15$$

Add \frac{22y}{25} to -\frac{9y}{10}.

$$-0.02y=-3.19+3a+3.15$$

Add -3.19 to 3.15+3a.

$$-0.02y=3a-0.04$$

Multiply both sides by -50.

$$y=2-150a$$

Substitute 2-150a for y in 11x+6y=-20a-21. Because the resulting equation contains only one variable, you can solve for x directly.

$$11x+6\left(2-150a\right)=-20a-21$$

Multiply 6 times 2-150a.

$$11x+12-900a=-20a-21$$

Subtract 12-900a from both sides of the equation.

$$11x=880a-33$$

Divide both sides by 11.

$$x=80a-3$$

The system is now solved.

$$x=80a-3,y=2-150a$$