2 Substitution Problems

[juicytomato]

These are problems I've been working on for a little while now. I'm not good at substitution, especially when there are fractions involved.

5x + y = 8
–4x – 4y = – 16

and

8/5x+3y
2/3x4/9y=-2

I'm feeling really stupid now, I know these are easy problems. I just have trouble getting started, I guess.

 

[tkhunny]

5x + y = 8
–4x – 4y = – 16

One thing that you may not have seen is simplifying what you have. This one is begging for division by -4

5x + y = 8 is OK, but
-4x - 4y = -16 can be changed to x + y = 4

Substitution, then, is a matter of:
1) Solve for something. Anything. No one cares.
y = 8 -5x
2) Substitute into the equation you didn't pick in 1).
x + y = 4
x + (8-5x) = 4
3) Solve for what you have left.
4) Substitute into ANY previous equation to find the other.
5) Check your results in ORIGINAL equations.

8/5x+3y
2/3x4/9y=-2

This one has a few problems. You are missing a few symbols.

Anyway, don't let the fractions scare you. Fractions are our friends. Just send them on a long vacation.

For example:
(8/5)*x + 3y = 1/10

Interesting Denominators are 5 and 10. The LCD is 10. Multiply by 10.

10 *[(8/5)*x + 3y] = 10*[1/10]
10 *(8/5)*x + 10*3y = 10*(1/10)
2 *8*x + 30y = 1
16x + 30y = 1 <== Where did the fractions go?

Similarly,
(2/3)x + (4/9)y=-2

We can simplify by dividing by 2
(1/3)x + (2/9)y=-1

We can clear fractions by multiplying by the LCD, 9
3x + 2y=-9

Do these look any easier?
16x + 30y = 1
3x + 2y=-9

Of course, I made up these equations, so they won't lead to the correct answers on this problem.

 

[wjm11]

5x + y = 8
–4x – 4y = – 16

Two different approaches:

1) Multiply the first eqn by 4, then add it to the second eqn. Why? The second eqn has a –4y in it, so the y terms will add up to zero. Then it’s easy to solve for x. After finding x, substitute it back into one of the eqns to find y. Finally, check your results by plugging both x and y values into the other eqn to see if they work there.

2) Rearrange the first eqn to solve for y:

y = 8 – 5x

Plug this into the second eqn for y:

-4x – 4(8 – 5x) = -16

Solve for x, then continue as in approach 1).

 

[wjm11]

TK, you're too quick for me!