Find the point of intersection of the pair of straight lines 6x + 4y = 13 and −7x + 7y = 14.

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The equations indeed represent two straight lines, because they are both linear (have the form `A x + B y = C` ). To find the point of intersection is the same as to solve the system of these equations, `{(6 x + 4 y = 13),(-7 x + 7 y = 14):}` It is because the pair `( x , y )` satisfies an equation is the same as the point `( x , y )` lies on the corresponding line.

There are several means to solving system of two linear equations with two unknowns. Using any of them, to reduce our work, divide both parts of the second equation by 7 to obtain an equivalent equation `- x + y = 2.`

Now we can easily express y in terms of x: `y = 2 + x.` Substitute this expression to the first equation:

`6 x + 4 ( 2 + x ) = 13,` which is the same as `10 x = 13 - 8 = 5.` Now we see `x = 5 / 10 = 1 / 2.`

Finally, find `y = 2 + x = 5 / 2.` The point we had to find is `( 1 / 2, 5 / 2 ) .`

We can check the solution: `6 x + 4 y = 3 + 10 = 13` (true), `- 1 / 2 + 5 / 2 = 4/2 = 2` (true).

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