Question

How do you solve this system of equations: `6x - 7y = 19 and 3x + 4y = 2`?

Answer 1

See a solution process below:

Explanation:

Step 1) Solve each equation for `6x`;

• Equation 1:

`6x - 7y = 19`

`6x - 7y + color(red)(7y) = 19 + color(red)(7y)`

`6x - 0 = 19 + 7y`

`6x = 19 + 7y`

• Equation 2:

`3x + 4y = 2`

`color(red)(2)(3x + 4y) = color(red)(2) xx 2`

`(color(red)(2) xx 3x) + (color(red)(2) xx 4y) = 4`

`6x + 8y = 4`

`6x + 8y - color(red)(8y) = 4 - color(red)(8y)`

`6x + 0 = 4 - 8y`

`6x = 4 - 8y`

Step 2) Because the left side of both equations are equal we can equate the right sides of both equations and solve for `y`:

`19 + 7y = 4 - 8y`

`19 - color(red)(19) + 7y + color(blue)(8y) = 4 - color(red)(19) - 8y + color(blue)(8y)`

`0 + (7 + color(blue)(8))y = -15 - 0`

`15y = -15`

`(15y)/color(red)(15) = -15/color(red)(15)`

`(color(red)(cancel(color(black)(15)))y)/cancel(color(red)(15)) = -1`

`y = -1`

Step 3) Substitute `-1` for `y` in the solution to either equation in Step 1 and solve for `x`:

`6x = 19 + 7y` becomes:

`6x = 19 + (7 xx -1)`

`6x = 19 + (-7)`

`6x = 19 - 7`

`6x = 12`

`(6x)/color(red)(6) = 12/color(red)(6)`

`(color(red)(cancel(color(black)(6)))x)/cancel(color(red)(6)) = 2`

`x = 2`

The Solution Is:

`x = 2` and `y = -1`

Or

`(2, -1)`

Answer 2

`x=2`; `y=-1`

Explanation:

Let `6x−7y=19` be equation 1,
and `3x+4y=2` be equation 2.

Since one of the coefficients of the variables is a multiple of the coefficient of the same variable in the other equation, the system can be solved by elimination.

To do this, `-2` will be multiplied to all the terms in the second equation so that `6x` can be canceled out from the first one once combined. Rewriting the two equations:

`6x−7y=19`
`-6x-8y=-4`

Upon combining the two equations, the result would be

`cancel(6x)−7y=19`
`cancel(-6x)-8y=-4`

`-7y-8y=19-4`
`-15y=15`

With `y=-1`.

To find for `x`, substitute y in any of the two equations. For this example, equation 1 was used.

`6x−7y=19`
`6x−7(-1)=19`
`6x+7=19`
`6x=12`

Finally, `x=2`.

Source: https://www.mathportal.org/algebra/solving-system-of-linear-equations/elimination-method.php