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

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)#

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