Step by step: way 1, elimination
The question
Question: solve 3x + 2y = 12 and 5x − 2y = 4.
3x + 2y = 12①
We call them ① and ②.
We want the x and y that make both true.
No letter is alone here.
So we add the equations to remove one letter.
This way is called elimination.
5x − 2y = 4②
1 Add the equations
Look at y: 2y and −2y.
They are opposites.
Added, they make zero.
2y and −2y make zero.
Cross them out.
We add ① and ②: the x parts together, and the numbers together.
3x + 5x = 8x.
12 + 4 = 16.
3x + 5x = 8x
12 + 4 = 16
Now we have 8x = 16.
8x = 16
Is x alone now?
Not yet: 8x means 8 times x.
How do we remove it?
2 Find x
To remove times 8, we divide by 8.
We do it on both sides, so they stay equal.
8x8 = 168
8 divided by 8 is 1.
They cancel each other.
So just x is left.
16 divided by 8 is 2.
168 = 2
So x = 2.
x = 2
So after step 2: we divided by 8, and now we have x = 2.
← step 2: divided by 8
3 Find y
We put x = 2 into ①.
3 times 2 is 6.
3(2) + 2y = 12
Now we have 6 + 2y = 12.
6 + 2y = 12
We want y by itself.
The 6 is in the way.
Is y alone now?
Not yet: + 6 is still with y.
To remove 6, we take away 6.
We do it on both sides, so they stay equal.
6 + 2y − 6 = 12 − 6
6 and − 6 cancel each other.
They make zero.
12 minus 6 is 6.
12 − 6 = 6
Now we have 2y = 6.
2y = 6
2y means 2 times y.
Is y alone now?
Not yet: 2y means 2 times y.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2y2 = 62
2 divided by 2 is 1.
They cancel each other.
So just y is left.
6 divided by 2 is 3.
62 = 3
So y = 3.
y = 3
So x = 2 and y = 3, both at the same time.
We write it as (2, 3).
On a graph, it is the point where the two lines meet.
(x, y) = (2, 3)
4 Check
Let's check in both equations.
We put x = 2 and y = 3 into ①.
① 3(2) + 2(3) = 6 + 6 = 12
We get 12, the same as the right side.
Now ②.
② 5(2) − 2(3) = 10 − 6 = 4
We get 4, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: add the equations.
Then: find x.
Then: find y.
Last: check.
Step by step: way 2, graphing
The question
Question: solve by graphing 3x + 2y = 12 and 5x − 2y = 4.
① 3x + 2y = 12 and ② 5x − 2y = 4
The answer is a point that is on both lines.
So we draw both lines and see where they meet.
answer = the point on both lines
1 Crossing points of ①
Line ① is not y = mx + b.
So we find where it crosses the two axes.
Two points are enough for a line.
On the x-axis, y = 0.
Put 0 in place of y.
3x + 2 × 0 = 12
2 times 0 is 0.
Adding 0 changes nothing.
3x = 12
3x means 3 times x.
Is x alone now?
Not yet: 3x means 3 times x.
How do we remove it?
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3x3 = 123
3 divided by 3 is 1.
They cancel each other.
So just x is left.
12 divided by 3 is 4.
123 = 4
So x = 4.
x = 4
On the y-axis, x = 0.
Put 0 in place of x.
3 × 0 + 2y = 12
3 times 0 is 0.
0 plus 2y is just 2y.
2y = 12
2y means 2 times y.
Is y alone now?
Not yet: 2y means 2 times y.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2y2 = 122
2 divided by 2 is 1.
They cancel each other.
So just y is left.
12 divided by 2 is 6.
122 = 6
So y = 6.
So line ① crosses the axes at (4, 0) and (0, 6).
y = 6
So after step 1: we divided by 3 and divided by 2, and now we have y = 6.
← step 1: divided by 3 and divided by 2
2 Crossing points of ②
Line ② is not y = mx + b.
So we find where it crosses the two axes.
Two points are enough for a line.
On the x-axis, y = 0.
Put 0 in place of y.
5x − 2 × 0 = 4
2 times 0 is 0.
Taking away 0 changes nothing.
5x = 4
5x means 5 times x.
Is x alone now?
Not yet: 5x means 5 times x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5x5 = 45
5 divided by 5 is 1.
They cancel each other.
So just x is left.
So x = 4/5.
x = 45
On the y-axis, x = 0.
Put 0 in place of x.
5 × 0 − 2y = 4
5 times 0 is 0.
0 minus 2y is −2y.
−2y = 4
−2y means −2 times y.
Is y alone now?
Not yet: −2y means −2 times y.
To remove times −2, we divide by −2.
We do it on both sides, so they stay equal.
−2y−2 = 4−2
−2 divided by −2 is 1.
They cancel each other.
So just y is left.
Plus over minus is minus.
4 divided by −2 is −2.
4−2 = −2
So y = −2.
So line ② crosses the axes at (0.8, 0) and (0, −2).
y = −2
So after step 2: we divided by 5 and divided by 2, and now we have y = −2.
← step 2: divided by 5 and divided by 2
3 Graph line ①
Draw the grid: the x-axis goes across, and the y-axis goes up and down.
They meet at 0, a point called the origin.
Put the two crossing points on the grid,
and join them.
4 Graph line ②
Put the two crossing points on the grid,
and join them.
5 Where they meet
The lines meet at one point.
That point is on both lines, so it makes both equations true.
Go straight down to the x-axis: x = 2.
Go straight across to the y-axis: y = 3.
So x = 2 and y = 3, both at the same time: the point (2, 3).
(x, y) = (2, 3)
6 Check in both
Let's check: put x = 2 and y = 3 into both equations.
First ①.
① 3(2) + 2(3) = 6 + 6 = 12
We get 12, the same as the right side.
It works.
Now ②.
② 5(2) − 2(3) = 10 − 6 = 4
We get 4, the same as the right side.
It works.
(2, 3) is on both lines.
So the answer is right!
Let's look back.
First: crossing points of ①.
Then: crossing points of ②.
Then: graph line ①.
Then: graph line ②.
Then: where they meet.
Last: check in both.
