Worked examples
The question
Question: y varies directly with x.
y = 12 when x = 4.
Find y when x = 7.
1 Write the rule
Varies directly means y is always the same number times x.
When x gets bigger, y gets bigger too.
We call the number k: y = kx.
y = kx
k stays the same for every pair of numbers.
Books call it the constant of variation.
k stays the same
2 Find k
We put in y = 12 and x = 4.
k times 4 is written 4k.
12 = 4k
4k means 4 times k.
We want just k.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
Careful!
k is y divided by x: 12/4.
Not 4/12.
The y number goes on top.
Careful: not k = 412 ✗
12/4 on the left.
4k/4 on the right.
124 = 4k4
4 divided by 4 is 1.
So just k is left.
12 divided by 4 is 3.
124 = 3
So k = 3.
k = 3
3 Write the equation
We put 3 in place of k.
So the equation is y = 3x.
y = 3x
4 Find y
We put in x = 7.
y = 3 × 7
3 times 7 is 21.
3 × 7 = 21
So y = 21.
y = 21
5 Check
Let's check.
In direct variation, y divided by x is always k.
First pair: 12/4 is 3.
y ÷ x: 124 = 3
New pair: 21/7 is 3 too.
The same k.
So the answer is right!
217 = 3
Let's look back.
First: write the rule.
Then: find k.
Then: write the equation.
Then: find y.
Last: check.
▶ Watch it solved
The question
Question: y varies directly with x.
y = 14 when x = 4.
Find y when x = 6.
1 Write the rule
Varies directly means y is always the same number times x.
When x gets bigger, y gets bigger too.
We call the number k: y = kx.
y = kx
k stays the same for every pair of numbers.
Books call it the constant of variation.
k stays the same
2 Find k
We put in y = 14 and x = 4.
k times 4 is written 4k.
14 = 4k
4k means 4 times k.
We want just k.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
Careful!
k is y divided by x: 14/4.
Not 4/14.
The y number goes on top.
Careful: not k = 414 ✗
14/4 on the left.
4k/4 on the right.
144 = 4k4
4 divided by 4 is 1.
So just k is left.
We can make 14/4 simpler.
It is the same as 7/2.
144 = 72
So k = 7/2.
k = 72
3 Write the equation
We put 7/2 in place of k.
So the equation is y = 7/2 times x.
y = 72x
4 Find y
We put in x = 6.
y = 72 × 6
7/2 times 6 is 21.
72 × 6 = 21
So y = 21.
y = 21
5 Check
Let's check.
In direct variation, y divided by x is always k.
First pair: 14/4 is 7/2.
y ÷ x: 144 = 72
New pair: 21/6 is 7/2 too.
The same k.
So the answer is right!
216 = 72
Let's look back.
First: write the rule.
Then: find k.
Then: write the equation.
Then: find y.
Last: check.
▶ Watch it solved
The question
Question: y varies directly with the square of x.
y = 50 when x = 5.
Find y when x = 3.
1 Write the rule
Varies directly means y is always the same number times x².
When x gets bigger, y gets bigger too.
We call the number k: y = kx².
y = kx2
k stays the same for every pair of numbers.
Books call it the constant of variation.
k stays the same
2 Find k
We put in y = 50 and x = 5.
5 squared is 25, and k times 25 is written 25k.
50 = 25k
5² = 5 × 5 = 25
25k means 25 times k.
We want just k.
To remove times 25, we divide by 25.
We do it on both sides, so they stay equal.
Careful!
k is y divided by x²: 50/25.
Not 25/50.
The y number goes on top.
Careful: not k = 2550 ✗
50/25 on the left.
25k/25 on the right.
5025 = 25k25
25 divided by 25 is 1.
So just k is left.
50 divided by 25 is 2.
5025 = 2
So k = 2.
k = 2
3 Write the equation
We put 2 in place of k.
So the equation is y = 2x².
y = 2x2
4 Find y
We put in x = 3.
y = 2 × 32
3 squared is 9.
3² = 9
2 times 9 is 18.
2 × 9 = 18
So y = 18.
y = 18
5 Check
Let's check.
In direct variation, y divided by x² is always k.
First pair: 50/25 is 2.
y ÷ x²: 5025 = 2
New pair: 18/9 is 2 too.
The same k.
So the answer is right!
189 = 2
Let's look back.
First: write the rule.
Then: find k.
Then: write the equation.
Then: find y.
Last: check.
▶ Watch it solved
