Question

Solving System of Linear Equations

Lesson Posttest
Question 2 Points 2
Find the solutions to the system of equations 4x - 3y = 11 and 2x - 7y= 22.
O (0.5, -3)
O (0.5, 3)
O (-0.5, -3)

Answer 1

(0.5, -3)

Explanation:

I'll use substitution and elimination to show two different ways of solving the equation.

Substitution:

So we can either substitute x or y and it really doesn't matter. It may be more simple to substitute a certain variable in some equations.

4x - 3y = 11

4x = 3y + 11

2x = (3y + 11) / 2

In this case I didn't completely solve for x since I only need the value of 2x to substitute into the second equation

`(3y+11)/(2)-7y=22\\\\2((3y+11)/(2)-7y)=2(22)\\3y+11-14y=44\\11-11y=44\\-11y=33\\y=-3`

Now that we have the value y we can substitute it back into either equation to solve for x.

`4x-3(-3) = 11\\4x+9=11\\4x=2\\x=0.5`

So the solution is (0.5, -3)

Elimination:

In this method we manipulate one of the equations to get the absolute value of one of the coefficients equal to the coefficient in the other equation with the only difference being the sign, so one is positive and one is negative. It doesn't matter which variable you chose to cancel out nor does it matter which equation you manipulate but it can sometimes be more simple to try to cancel out a certain variable if the value is smaller.

`4x-3y = 11\\-0.5(4x-3y) = -0.5(11)\\-2x+1.5y=-5.5\\`

Now that we have an equation where the -2x can be canceled out by adding the other equation 2x-7y = 22 we add the two equations to get

`\ \ \ -2x + 1.5y = -5.5\\+2x - 7y = 22\\ 0 -5.5y = 16.5\\`

Now that you canceled out the x you can solve for y and then use substitution to solve for x

`-5.5y = 16.5\\y=-3\\`

Now substitute this into any of the original equations to solve for x

`2x - 7(-3) = 22\\\2x+21=22\\2x=1\\x=1/2`

So the solution is (0.5, -3)