Question

Choose a method to solve the linear system. Explain your choice. $$ \begin{aligned} &6.2 x-0.5 y=-27.8\\\ &0.3 x+0.4 y=68.7 \end{aligned} $$

Short answer

Solution to the system of linear equations is \(x \approx 8.83, y \approx 165.12 \)

Step-by-step solution

Multiply the equations

To apply the elimination method, one of the equations should be multiplied by a suitable factor so that the coefficients of y's (or x's) in both equations will cancel each other when subtracted. In this case, let's multiply the first equation by 0.4 and the second by 0.5 to cancel out the y's coefficients. Hence, the system of equations becomes: \[ \begin{aligned} &2.48 x - 0.2 y = -11.12\ &0.15 x + 0.2 y = 34.35 \end{aligned} \]

Add the two equations

Adding both equations results in an equation with only one variable: \[ 2.63x = 23.23 \]

Solve for X

Solving for x involves dividing both sides of the equation by the coefficient of x, which gives: \( x = 23.23 / 2.63\), which evaluates to \(x \approx 8.83\)

Substitute x into one of the original equations

Substitute X into the second equation and solve for y. This gives: \[ 0.3*8.83 + 0.4y = 68.7 \] Simplifying gives: \(0.4y = 68.7 - 2.65 = 66.05\) hence, \(y = 66.05 / 0.4 \approx 165.12 \)