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Camp · A line that starts somewhere
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Start with the main idea.
1 of 10 · linear-mxb-c01
y = 4x + 7. When x = 3, y = ?
The 4 is the rate: each time x goes up by 1, y goes up by 4.
The 7 is the start, the amount already there when x is 0.
Put 3 in for x. 4 times 3 is 12, and 12 plus 7 is 19.
So y is 19.
Check: 4×3+7=19, multiplying before adding.
Answer: 19
2 of 10 · linear-mxb-c02
y = 5x + 2. What is b, the start?
In y = mx + b, b is the number added on at the end, not the one next to x.
This equation is y = 5x + 2, so the added-on number is 2.
That 2 is what y equals when x is 0, before the rate starts repeating.
The 5 is the rate, so it is not the start.
So b is 2.
Check: At x=0, 5×0+2=2, the starting value.
Answer: 2
3 of 10 · linear-mxb-c03
y = 5x + 2. What is m, the rate?
In y = mx + b, m is the number multiplying x.
That number tells how much y changes every time x goes up by 1.
In y = 5x + 2, 5 multiplies x, so m = 5.
The 2 is the start, so leave it out of m.
So m is 5.
Check: At x=0 y=2 and at x=1 y=7; the increase is 5.
Answer: 5
4 of 10 · linear-mxb-c04
A table: x = 0,1,2 and y = 8,11,14. What is m?
Table. Columns: x, y. Rows: 0, 8; 1, 11; 2, 14.
x
y
0
8
1
11
2
14
m is the jump in y when x jumps by 1.
From the first row to the second, x goes from 0 to 1 and y goes from 8 to 11.
11 minus 8 is 3.
So m is 3.
Check: The next jump, 14−11, also equals 3.
Answer: 3
5 of 10 · linear-mxb-c05
This table follows a linear relationship. What is b, the value of y when x = 0?
Table. Columns: x, y. Rows: 0, 8; 1, 11; 2, 14.
x
y
0
8
1
11
2
14
We need b, the y-value when x is 0. The table rows are (0, 8), (1, 11), and (2, 14).
b is the start: the y-value when x is still 0.
The row x = 0, y = 8 is in the table, so b = 8.
The rows 11 and 14 belong to x = 1 and x = 2, so they are not the start.
Check: the question asks for y when x is 0, and that table cell is 8.
Answer: 8
6 of 10 · linear-mxb-c06
Is y = 4x + 7 proportional? yes or no
A proportional relationship has a straight-line graph through (0, 0), so its starting value b is 0.
This equation is y = 4x + 7, so the start is 7, not 0.
The 4 is a steady rate, so the graph is still a straight line.
A straight line with a nonzero starting value is linear but not proportional.
So the answer is no.
Check: At x = 0, y = 7, so the line misses the origin, the point (0, 0).
Answer: no
7 of 10 · linear-mxb-c07
Is y = 4x proportional? yes or no
Straight-line graph. Horizontal axis x, from 0 to 5. Vertical axis y, from 0 to 20. Marked points: (0, 0); (2, 8); (4, 16). The line passes through the origin, the point (0, 0).
y = 4x means y = 4x + 0, so the start b is 0.
When x is 0, y is also 0, and the graph goes through (0, 0).
The 4 says y is always 4 times x, which is one steady rate.
A straight line through the origin, (0, 0), represents a proportional relationship.
So the answer is yes.
Check: At x=2 and 4, y/x is 8/2=4 and 16/4=4.
Answer: yes
8 of 10 · linear-mxb-c08
y = −2x + 9. When x = 4, y = ?
The -2 means each time x goes up by 1, y goes down by 2.
The 9 is the start, the value when x is 0.
Put 4 in for x: -2 times 4 is -8.
Then -8 plus 9 is 1.
So y is 1.
Check: Four decreases of 2 use up 8 of the starting 9, leaving 1.
Answer: 1
9 of 10 · linear-mxb-c09
A gym: $20 start plus $8 per class. Cost of 6 classes?
The $20 is the one-time join fee, so that is the start b.
The $8 is the charge for each class, so that is the rate m.
Six classes means x is 6.
8 times 6 is 48, and 48 plus the 20 start fee is 68.
So the cost is 68 dollars.
Check: Six $8 classes cost $48; $68−$48=$20, the join fee.
Answer: 68
10 of 10 · linear-mxb-c10
A graph crosses the y-axis at 4 and goes up 2 for every 1 to the right. b = ?
Straight-line graph. Horizontal axis x, from 0 to 5. Vertical axis y, from 0 to 16. Marked points: (0, 4); (1, 6); (3, 10).
b is where the line crosses the y-axis, the y-value when x is 0.
The graph is marked at (0, 4), so it crosses at 4.
Going up 2 for every 1 to the right is the slope m, not the start.
You can read b from the crossing point without using the slope.
So b is 4.
Check: Starting at y=4, one step right adds 2 and reaches 6.
Answer: 4
A line that starts somewhere · Camp
1 of 10 · linear-mxb-c01
y = 4x + 7. When x = 3, y = ?
The 4 is the rate: each time x goes up by 1, y goes up by 4.
The 7 is the start, the amount already there when x is 0.
Put 3 in for x. 4 times 3 is 12, and 12 plus 7 is 19.
So y is 19.
Check: 4×3+7=19, multiplying before adding.
Answer: 19
2 of 10 · linear-mxb-c02
y = 5x + 2. What is b, the start?
In y = mx + b, b is the number added on at the end, not the one next to x.
This equation is y = 5x + 2, so the added-on number is 2.
That 2 is what y equals when x is 0, before the rate starts repeating.
The 5 is the rate, so it is not the start.
So b is 2.
Check: At x=0, 5×0+2=2, the starting value.
Answer: 2
3 of 10 · linear-mxb-c03
y = 5x + 2. What is m, the rate?
In y = mx + b, m is the number multiplying x.
That number tells how much y changes every time x goes up by 1.
In y = 5x + 2, 5 multiplies x, so m = 5.
The 2 is the start, so leave it out of m.
So m is 5.
Check: At x=0 y=2 and at x=1 y=7; the increase is 5.
Answer: 5
4 of 10 · linear-mxb-c04
A table: x = 0,1,2 and y = 8,11,14. What is m?
Table. Columns: x, y. Rows: 0, 8; 1, 11; 2, 14.
x
y
0
8
1
11
2
14
m is the jump in y when x jumps by 1.
From the first row to the second, x goes from 0 to 1 and y goes from 8 to 11.
11 minus 8 is 3.
So m is 3.
Check: The next jump, 14−11, also equals 3.
Answer: 3
5 of 10 · linear-mxb-c05
This table follows a linear relationship. What is b, the value of y when x = 0?
Table. Columns: x, y. Rows: 0, 8; 1, 11; 2, 14.
x
y
0
8
1
11
2
14
We need b, the y-value when x is 0. The table rows are (0, 8), (1, 11), and (2, 14).
b is the start: the y-value when x is still 0.
The row x = 0, y = 8 is in the table, so b = 8.
The rows 11 and 14 belong to x = 1 and x = 2, so they are not the start.
Check: the question asks for y when x is 0, and that table cell is 8.
Answer: 8
6 of 10 · linear-mxb-c06
Is y = 4x + 7 proportional? yes or no
A proportional relationship has a straight-line graph through (0, 0), so its starting value b is 0.
This equation is y = 4x + 7, so the start is 7, not 0.
The 4 is a steady rate, so the graph is still a straight line.
A straight line with a nonzero starting value is linear but not proportional.
So the answer is no.
Check: At x = 0, y = 7, so the line misses the origin, the point (0, 0).
Answer: no
7 of 10 · linear-mxb-c07
Is y = 4x proportional? yes or no
Straight-line graph. Horizontal axis x, from 0 to 5. Vertical axis y, from 0 to 20. Marked points: (0, 0); (2, 8); (4, 16). The line passes through the origin, the point (0, 0).
y = 4x means y = 4x + 0, so the start b is 0.
When x is 0, y is also 0, and the graph goes through (0, 0).
The 4 says y is always 4 times x, which is one steady rate.
A straight line through the origin, (0, 0), represents a proportional relationship.
So the answer is yes.
Check: At x=2 and 4, y/x is 8/2=4 and 16/4=4.
Answer: yes
8 of 10 · linear-mxb-c08
y = −2x + 9. When x = 4, y = ?
The -2 means each time x goes up by 1, y goes down by 2.
The 9 is the start, the value when x is 0.
Put 4 in for x: -2 times 4 is -8.
Then -8 plus 9 is 1.
So y is 1.
Check: Four decreases of 2 use up 8 of the starting 9, leaving 1.
Answer: 1
9 of 10 · linear-mxb-c09
A gym: $20 start plus $8 per class. Cost of 6 classes?
The $20 is the one-time join fee, so that is the start b.
The $8 is the charge for each class, so that is the rate m.
Six classes means x is 6.
8 times 6 is 48, and 48 plus the 20 start fee is 68.
So the cost is 68 dollars.
Check: Six $8 classes cost $48; $68−$48=$20, the join fee.
Answer: 68
10 of 10 · linear-mxb-c10
A graph crosses the y-axis at 4 and goes up 2 for every 1 to the right. b = ?
Straight-line graph. Horizontal axis x, from 0 to 5. Vertical axis y, from 0 to 16. Marked points: (0, 4); (1, 6); (3, 10).
b is where the line crosses the y-axis, the y-value when x is 0.
The graph is marked at (0, 4), so it crosses at 4.
Going up 2 for every 1 to the right is the slope m, not the start.
You can read b from the crossing point without using the slope.
So b is 4.
Check: Starting at y=4, one step right adds 2 and reaches 6.
