Substitution isn't too difficult to grasp. What you must first come to understand is that in both equations (in this particular problem) x and y are assumed to be the same numbers respectively. Meaning if you looked at the x value of the function: (Keep in mind [tex] \frac{x}{2} [/tex] is the same as [tex] \frac{1}{2} x[/tex])
[tex] y= \frac{x}{2} + 7 [/tex]
It would be the same as the x value in the function:
[tex]-x = 3y - 11[/tex]
Due to having too many variables we can't solve for a specific variable with the equations in there current form, but because they have equal values of x and y we can solve for x or y and SUBSTITUTE that value for x or y into the other equation! By doing so we eliminate a variable and make it possible to solve that equation for one of the two variables. Luckily they already solved the equations for y in the first one (Keep in mind x, and -x are not the same, thus if you want to use the -x equation you'll need to multiply all numbers by -1.
Explanation aside, let's get to the problem!
[tex] y= \frac{x}{2} + 7 [/tex]
Now we take that value for y, and substitute it in the other equation wherever there is a y.
[tex]-x = 3[tex]( \frac{x}{2}+7)[/tex]- 11[/tex]
Distribute the 3 to all terms in the parenthesis. (Remember only multiply the numerator by 3! (don't multiply the denominator by 3 is what I'm trying to say)... also if you have forgotten.. the numerator is the top of the fraction and the denominator is the bottom!)
[tex]-x= \frac{3x}{2}+21 - 11 [/tex]
Subtract 11 from 21.
[tex]-x= \frac{3x}{2}+10[/tex]
Subtract 10 on both sides.
[tex]-10-x= \frac{3x}{2}+10-10[/tex]
Simplify the right side (10-10 = 0) and add x on both sides!!
[tex]-10 = \frac{3x}{2} +x[/tex]
Add the x's together. To do this gain a common denominator by multiplying the +x by 2/2. Then add it together!
[tex]-10 = \frac{5x}{2} [/tex]
To get x alone we need to remove the fraction. To do that multiply by the fractions reciprocal (flip the fraction and multiply both sides by it. Keep in mind you do not want to multiply by x in this case[tex] \frac{2}{5}*-10 = \frac{5x}{2} * \frac{2}{5} [/tex])
Cancel out the fraction on the right side, and multiply the -10 by 2/5. [tex] \frac{-20}{5} = x[/tex]
x = -4
You probably are thinking... wow that's a lot of work! Well you are somewhat right, but luckily the next part is pretty easy. Pick one of the equations, and plug in the value you got for x and solve for the other variable (y). I'm going to choose the one with y already solved for.
[tex] \frac{-4}{2} + 7 = y [/tex]
[tex]y = -2 + 7 [/tex]
[tex]y = 5[/tex]
So our answers are:
x = -4
y = 5
