Simultaneous equations by SUBSTITUTION?

Question

5x - 2y = 10
3x + 2y = 6

Please can someone advise how to do this. I DONT JUST WANT AN ANSWER. I wanna understand how it was done.

Answer 1

Okay, the goal for solving simultaneous equations is to isolate one variable, so rearrange one of the equations (it doesn't matter which one) until you get either x or y alone on one side.

5x - 2y = 10
5x - 2y + 2y = 10 + 2y (if you do something to one side, do it to the other)
5x = 10 + 2y
5x/5 = (10 + 2y) / 5
x = (10+2y)/5

Then, plug this new value for x (the right side of the equation) into the other equation, hence the "substitution."
3x + 2y = 6
3 ((10+2y)/5) + 2y = 6
(30 + 60y)/5 + 2y = 6
6 + 12y + 2y = 6
14y = 0
y = 0

Now, solve for x by substituting this y value back into one of the equations.

5x - 2y = 10
5x - 2(0) = 10
5x = 10
x=2

So, x=2 and y=0

Answer 2

The idea of substitution is to get an equation in which one of the two variables is "defined in terms of" the other. This is much easier to do than to explain, so I'll go ahead and do it:

5x - 2y = 10. The first step is to get the two variables to separate sides of the equation, so add 2y to each side.

5x = 10 + 2y. The next thing to do is get that x alone, so divide both sides by five.

x = 2 + 2y/5. Now, you have an equation in terms of y that is equivalent to x. Next comes the substitution part. Go back to the other equation, 3x + 2y = 6. Now, wherever x appears in that equation, substitute in the equation you just found.

3(2 + 2y/5) + 2y = 6. Now you have an equation with just one variable. Distribute out the 3: 6 + 6y/5 + 2y = 6. Subtract 6 from both sides and you get 6y/5 + 2y = 0. At this point there's no need to combine the y terms because y must be 0.

Now go back to the equation you derived first--x = 2 + 2y/5. Since y is 0, that simplifies to x = 2.

Finally, check your work. 2*5 = 10 and 2*3 = 6, so it checks out.

I hope you understand better how to solve these problems now.

Answer 3

make "y" the subject of one of the equations and replace the expression you get into the "y" of the other equation or you can do the same thing but with the "x" depending on what is easier for you
5x - 2y = 10
2y = 5x -10
y = 2.5x -5

3x + 2(2.5x-5) = 6
3x + 5x -10 = 6
8x = 16
x = 2
then replace "x" in the equation with "y" being the subject
y = 2.5(2) - 5
y = 0

Answer 4

first you need to get the equations in a useful form:
for 5x-2y=10:
subtract 5x from both sides: -2y=10-5x
divide by -2: y=5/2x-5

for 3x+2y=6:
subtract 3x from both sides: 2y=6-3x
divide by 2: y=3-3/2x

so now you have:
y=5/2x-5 and y=3-3/2x

now the substitution:
if y=5/2x-5 AND y=3-3/2x you can replace y in one of the equations with the other equation
so I chose to substitute the 'y' from the second equation with 5/2x-5

now you have 5/2x-5=3-3/2x
solve for x.
add 5 to both sides: 5/2x=8-3/2x
add 3/2x to both sides: 8/2x=8 or 4x=8
divide by 4: x=2

Answer 5

the easy way to do this is add the
two equations together
8x=16>>>>x=2
and y=0

substitution
5x - 2y = 10.........(1)
3x + 2y = 6..........(2)

take 3x from each side of (2)
2y=6-3x
now, substitute this y into (1)
5x-(6-3x)=10
5x-(-3x)-6=10
add 6 to each side and
drop brackets
8x-6+6=10+6
8x=16 >>>>x=2
substitute x value into (2)
6+2y=6
2y=6-6=0
y=0

ihope that this helps

Answer 6

if 3x + 2y =6, then 2y = 6-3x.

substitute that 2y into the first equation:
5x-(6-3x) = 10
5x - 6 + 3x =10
8x = 16
x=2
and y=0

Answer 7

5x - 2y = 10
3x + 2y = 6

add the equations together

8x = 16
x = 2

5x - 2y = 10
10 - 2y = 10
-2y = 0
y = 0

x = 2
y = 0