Gradients and y = mx + c

In the equation y=mx+c, the gradient is m.

Rewrite the equation with the x term first. Keep the sign that goes with it.

Every point on the y-axis has an x-coordinate of 0.

Choose two points where the line passes exactly through a corner of the grid. Find how far up you go for each 1 across.

Does the line go up or down as you move to the right? That tells you the sign of the gradient.

Gradient is the change in y divided by the change in x. Take the coordinates in the same order both times.

1. Change in y: 5-2=3
2. Change in x: 7-1=6
3. Gradient =\frac{\text{change in } y}{\text{change in } x}=\frac{3}{6}=\frac{1}{2}

How far does the line go up for each 1 across?

1. The line is horizontal, so y stays the same as x increases.
2. The change in y is 0, so the gradient is \frac{0}{\text{change in } x}=0.

Use y=mx+c. Which number is m and which is c?

1. A straight line has an equation of the form y=mx+c, where m is the gradient and (0, c) is where it crosses the y-axis.
2. m=-4 and c=6, so the equation is y=-4x+6.

Parallel lines have the same gradient. Find the gradient of each line.

1. Parallel lines have the same gradient.
2. The gradient of y=4x-3 is 4.
3. y=4x+7 also has gradient 4, so it is parallel.

Get y on its own first. The gradient is the number in front of x when the equation starts y = \ldots

1. Divide every term by 2 to get the equation in the form y=mx+c.
2. y=3x+2
3. The gradient is 3.

Find where the line crosses the y-axis. Then use two points where the line passes exactly through a corner of the grid to find the gradient.

1. The line crosses the y-axis at (0, 1), so c=1.
2. The line passes through (0, 1) and (2, 2). It goes up 1 for every 2 across.
3. Gradient m=\frac{1}{2}
4. The equation is y=\frac{1}{2}x+1

Write y=-2x+c, then put in the coordinates of the point to find c.

1. The equation is y=-2x+c.
2. The point (3, 1) is on the line, so put x=3 and y=1 into it: 1=-2\times3+c
3. 1=-6+c, so c=7.
4. The equation is y=-2x+7

Work out the gradient first. Then put one of the points into y=mx+c to find c.

1. Gradient =\frac{15-3}{5-2}=\frac{12}{3}=4
2. So y=4x+c. Put in (2, 3): 3=4\times2+c
3. 3=8+c, so c=-5.
4. The equation is y=4x-5
5. Check with (5, 15): 4\times5-5=15

Gradient 2 means y goes up 2 for every 1 across. How far across is it from (2, 3) to (6, k)?

1. From (2, 3) to (6, k), x increases by 6-2=4.
2. The gradient is 2, so y increases by 2\times4=8.
3. k=3+8=11

Look at the y-coordinates of the two points. What does that tell you about the line?

1. Both points have y-coordinate -2.
2. So the line is horizontal, and every point on it has y-coordinate -2.
3. The equation is y=-2

Parallel lines have the same gradient. Use the point to find where line M crosses the y-axis.

1. Parallel lines have the same gradient, so line M has gradient 3.
2. So line M is y=3x+c. Put in (2, 9): 9=3\times2+c
3. 9=6+c, so c=3.
4. Line M is y=3x+3

Use the numbers on the axes, not the number of squares, to find how much y changes for each 1 across.

1. On the y-axis each small square is 5 units, but on the x-axis each small square is 1 unit.
2. The line crosses the y-axis at (0, 10), so c=10.
3. The line passes through (0, 10) and (2, 20). For 2 across, y goes up by 10.
4. Gradient m=\frac{10}{2}=5
5. The equation is y=5x+10

A gradient of -2 means y goes down by 2 for every 1 you move to the right.

Part (a)

1. From (1, 9) to (k, 1), y goes down by 9-1=8.
2. The gradient is -2, so y goes down 2 for every 1 across.
3. 8\div2=4, so x goes across by 4.
4. k=1+4=5

Part (b)

1. The line is y=-2x+c.
2. Put in (1, 9): 9=-2\times1+c
3. 9=-2+c, so c=11.
4. Check with (5, 1): -2\times5+11=1

Rearrange the equation of line Q so that it starts y = \ldots Its gradient must equal the gradient of line P.

1. Line P has gradient 4, so line Q must also have gradient 4.
2. Divide the equation of line Q by 3: y=\frac{k}{3}x+\frac{2}{3}
3. So \frac{k}{3}=4
4. k=12
5. Check: 3y=12x+2 gives y=4x+\frac{2}{3}, which has gradient 4.

For the midpoint, add the two x-coordinates and halve, then do the same for the y-coordinates.

Part (a)

1. x: \frac{-2+6}{2}=\frac{4}{2}=2
2. y: \frac{5+1}{2}=\frac{6}{2}=3
3. M is (2, 3)

Part (b)

1. The line is y=-3x+c.
2. Put in M (2, 3): 3=-3\times2+c
3. 3=-6+c, so c=9.
4. The equation is y=-3x+9