{0,4 x+0,1 y=1 -0,1 x-0,4 y=-1.

To solve the system of linear equations:<br /><br />\[<br />\left\{<br />\begin{array}{l}<br />0.4x + 0.1y = 1 \\<br />-0.1x - 0.4y = -1<br />\end{array}<br />\right.<br />\]<br /><br />we can use the method of substitution or elimination. Here, we'll use the elimination method.<br /><br />First, let's multiply the first equation by 10 to eliminate the decimals:<br /><br />\[<br />4x + y = 10<br />\]<br /><br />Next, let's multiply the second equation by 10 as well:<br /><br />\[<br />-x - 4y = -10<br />\]<br /><br />Now we have the system:<br /><br />\[<br />\left\{<br />\begin{array}{l}<br />4x + y = 10 \\<br />-x - 4y = -10<br />\end{array}<br />\right.<br />\]<br /><br />Next, we'll multiply the second equation by 4 to align the coefficients of \(y\):<br /><br />\[<br />-4x - 16y = -40<br />\]<br /><br />Now we have:<br /><br />\[<br />\left\{<br />\begin{array}{l}<br />4x + y = 10 \\<br />-4x - 16y = -40<br />\end{array}<br />\right.<br />\]<br /><br />Add the two equations together to eliminate \(x\):<br /><br />\[<br />(4x + y) + (-4x - 16y) = 10 + (-40)<br />\]<br /><br />This simplifies to:<br /><br />\[<br />-15y = -30<br />\]<br /><br />Solving for \(y\):<br /><br />\[<br />y = 2<br />\]<br /><br />Now substitute \(y = 2\) back into the first equation:<br /><br />\[<br />4x + 2 = 10<br />\]<br /><br />Solving for \(x\):<br /><br />\[<br />4x = 8 \implies x = 2<br />\]<br /><br />So the solution to the system of equations is:<br /><br />\[<br />x = 2, \quad y = 2<br />\]<br /><br />Thus, the solution is \((2, 2)\).