8. ) -0.7x+0.6y=1.3 0.5x-0.3y=-0.8

To solve the system of linear equations:<br /><br />\[<br />\left\{<br />\begin{array}{l}<br />-0.7x + 0.6y = 1.3 \\<br />0.5x - 0.3y = -0.8<br />\end{array}<br />\right.<br />\]<br /><br />we can use either the substitution method or the elimination method. Here, we will use the elimination method.<br /><br />First, let's multiply both equations by suitable numbers to make the coefficients of \( y \) (or \( x \)) equal in magnitude but opposite in sign. We will eliminate \( y \).<br /><br />Multiply the first equation by 0.5 and the second equation by 0.6:<br /><br />\[<br />\begin{array}{l}<br />0.5(-0.7x + 0.6y) = 0.5(1.3) \\<br />0.6(0.5x - 0.3y) = 0.6(-0.8)<br />\end{array}<br />\]<br /><br />This gives us:<br /><br />\[<br />\begin{array}{l}<br />-0.35x + 0.3y = 0.65 \\<br />0.3x - 0.18y = -0.48<br />\end{array}<br />\]<br /><br />Now, add these two equations together to eliminate \( y \):<br /><br />\[<br />(-0.35x + 0.3y) + (0.3x - 0.18y) = 0.65 + (-0.48)<br />\]<br /><br />Simplify:<br /><br />\[<br />-0.35x + 0.3x + 0.3y - 0.18y = 0.65 - 0.48<br />\]<br /><br />Combine like terms:<br /><br />\[<br />-0.05x + 0.12y = 0.17<br />\]<br /><br />Since we made an error in our initial approach, let's correct it by re-evaluating the elimination process. Instead, let's multiply the first equation by 3 and the second equation by 6 to eliminate \( y \):<br /><br />\[<br />\begin{array}{l}<br />3(-0.7x + 0.6y) = 3(1.3) \\<br />6(0.5x - 0.3y) = 6(-0.8)<br />\end{array}<br />\]<br /><br />This gives us:<br /><br />\[<br />\begin{array}{l}<br />-2.1x + 1.8y = 3.9 \\<br />3x - 1.8y = -4.8<br />\end{array}<br />\]<br /><br />Now, add these two equations together to eliminate \( y \):<br /><br />\[<br />(-2.1x + 1.8y) + (3x - 1.8y) = 3.9 + (-4.8)<br />\]<br /><br />Simplify:<br /><br />\[<br />-2.1x + 3x + 1.8y - 1.8y = 3.9 - 4.8<br />\]<br /><br />Combine like terms:<br /><br />\[<br />0.9x = -0.9<br />\]<br /><br />Solve for \( x \):<br /><br />\[<br />x = -1<br />\]<br /><br />Now substitute \( x = -1 \) back into one of the original equations to solve for \( y \). Using the second equation:<br /><br />\[<br />0.5(-1) - 0.3y = -0.8<br />\]<br /><br />Simplify:<br /><br />\[<br />-0.5 - 0.3y = -0.8<br />\]<br /><br />Add 0.5 to both sides:<br /><br />\[<br />-0.3y = -0.3<br />\]<br /><br />Solve for \( y \):<br /><br />\[<br />y = 1<br />\]<br /><br />Thus, the solution to the system of equations is:<br /><br />\[<br />x = -1, \quad y = 1<br />\]