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Why is the point of intersect the solution for a system of equations?

The solution to the system of equations {y=x+1y=3x\begin{cases}y=x+1 \\ y=3-x \end{cases} is the values of xx and yywhich solve both equations y=x+1 och y=3x y=x+1 \quad \text{ och } \quad y=3-x at the samt time. Every equation has an infinite number of solutions. The table shows a few pair a whole numbers which solve the first equation.

xxxx\phantom{xx} x \phantom{x} xyxx\phantom{x} y \phantom{xx}
11 22
22 33
33 44
\vdots \vdots

You could say that all these number pairs lie along the straight line y=x+1.y=x+1. Here are a few shown, but in reality there are an infinite number of them.

In the same way, you can represent all the solutions for the equation y=3xy=3-x by using points along the line y=3x.y=3-x. Where can you find the number pair that solves both equations? Right where the lines intersect each other!

If you try putting x=1x=1 and y=2y=2 into both equations, you will see that both agree.