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Do the three lines $x_{1}-4 x_{2}=1,2 x_{1}-x_{2}=-3,$ and $-x_{1}-3 x_{2}=4$ have a common point of intersection? Explain.

$x_{1}=-13 / 7$$x_{2}=-5 / 7$$(-13 / 7,-5 / 7)$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 1

Systems of Linear Equations

Introduction to Matrices

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01:32

In mathematics, the absolu…

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Do the three planes $x_{1}…

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The intersection of the li…

For the following exercise…

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Do the equations $y-2=3(x-…

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and this problem. We're given three lines and asked to see if they have a common point of intersection and if so, to find it. So the first thing that we're going to do is write these lines in, make its form, and when we do, we get the Matrix written above, and once we have that, we're going to use the one. Cancel out this too, and negative one and just see where that leaves us. And it turns out that when we do that, we actually have this matrix here, and it's really noticeable that the second and third rose are basically opposites of each other's. We have seven, the negative seven and negative 55 which means that by adding road to grow three well, actually, just get all zeros in that role when we can actually completely cancel out that row and therefore now we have essentially two equations anti variables. So we're just gonna finish solving this matrix and so now will solve for so that the second variable they're getting a 01 and also get the 10 in the top. So we get e identity matrix when once we have the identity matrix that will give us our final solution, which shows that these three lines do into sect. They intersect in the point, and the point is minus 18 over seven, minus five over seven.

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