3.05 Find and use the equation of a line
Consider the equation of a line y = mx + c.
Consider the following three linear equations with their graphs plotted on a number plane:
Equation 1: y = x + 1
Equation 2: y = 2 x + 1
Equation 3: y = 4 x + 1
What do all of the equations have in common?
What do all of the graphs have in common?
Describe all lines that have the form: y = m x + 1
Consider the following three linear equations with their graphs plotted on a number plane:
Equation 1: y = 2 x + 4
Equation 2: y = 2 x + 8
Equation 3: y = 2 x - 4
What do all of the equations have in common?
What do all of the graphs have in common.
Describe all lines that have the form: y = 2 x + c
Determine whether or not the gradients of the following pairs of equations are equal:
Consider the line plotted below:
State the the y-intercept.
State the gradient.
Consider the line plotted below:
By how much does the y-value change as the x-value increases by 1?
State the gradient of the line.
State whether or not the following lines have a y-intercept:
y = 2
5 x = 4 y
y = - 4 x + 1
x = 1
y = 3 x
Consider the following linear equations:
State the value of the gradient, m.
State the y-intecept, c.
y = - 2 x + 9
y = - 4 x - 8
y = 8 x + 6
y = - 1 - \dfrac{9 x}{2}
- 9 x + 9 y = 27
3 x - 10 y =- 2
Sketch the following lines on a number plane:
The line with a y-intercept of - 2 and gradient of - 3.
The line with a y-intercept of 3 and gradient of - \dfrac{3}{2}.
The line y = 2 x + 5.
The line y = \dfrac{1}{2} x - 1.
For each of the following equations:
Find the y-value of the y-intercept of the line.
Find the x-value of the x-intercept of the line.
Find the value of y when x = 3.
Sketch the line on a number plane.
For each of the following lines:
Find the y-coordinate of the y-intercept of the line.
Hence, write the equation of the line in gradient-intercept form.
Find the x-coordinate of the x-intercept of the line.
Sketch the line on a number plane.
A line has gradient \dfrac{4}{5} and passes through the point \left( - 10 , 4\right).
A line has gradient - 2 and passes through the point \left(3, - 8 \right).
A line has a gradient of - 3 and intercepts the y-axis at - 2.
Find the equation of the line in the form y = m x + c.
State whether the point \left( - 2 , 4 \right) lies on this line.
A line has a gradient of - 3 and cuts the y-axis at 8.
Find the equation of the line in the form y = m x + c.
State whether the point \left( 8, - 31 \right) lies on this line.
Find the equations of the following in the form y = mx + c :
A line that has the same gradient as y = 9 - 8 x and the same y-intercept as
y = - 5 x - 3.
A line whose gradient is 2 and crosses the y-axis at 5.
A line whose gradient is - 6 and crosses the y-axis at 9.
A line whose gradient is - 8 and crosses the y-axis at - 9.
A line whose gradient is - \dfrac{3}{4} and intercepts the y-axis at 3.
A line whose gradient is \dfrac{4}{3} and goes through the point \left(0, 3\right).
A line that has a gradient of - 2 and passes through \left( - 6 , - 3 \right)
A line whose gradient is 8 and goes through the point \left(0, - 4 \right).
A line whose gradient is 0 and goes through the point \left(0, \dfrac{2}{13}\right).
Consider the line plotted on the number plane.
State the values of:
The gradient, m.
The y-intercept, c.
Find the equation of the line in gradient-intercept form.
Find the value of y when x = 27.
Consider the line plotted on the number plane.
Find the equation of the line in gradient-intercept form.
Find the value of y when x = 50.
Consider the line plotted on the number plane.
Find the equation of the line in gradient-intercept form.
Find the value of y when x = 29.
Find the equations of the following lines in gradient-intercept form:
For the following lines passing through the given two points:
Hence, state the gradient of the line.
Find the equation of the line in the form y = m x + c.
\left(0, 2\right) and \left(2, 6\right)
\left(0, - 9 \right) and \left(5, 1\right)
\left(0, 2\right) and \left( - 7 , 44\right)
For each of the following linear equations:
Rewrite it in the form y = m x + c.
State the gradient of the line, m.
State the y-intercept of the line, c.
y = \dfrac{- 4 x + 16}{4}
9 x - y - 8 = 0.
y = 3 \left( 4 x - 3\right)
y = 6 \left( 3 x - 2\right)
3 x - 9 y - 27 = 0.
3 x - 4 y - 28 = 0
Consider the lines with the following equations:
Line A: 5 x + 3 y + 5 = 0
Line B: 7 x + 6 y - 3 = 0
Express the lines in the form y = m x + c.
State which line is steeper, A or B.
Determine which of the following lines are steeper: 2 x + 5 y - 5 = 0 or 4 x + 4 y + 1 = 0.
A straight line has gradient -1 and goes through the points \left(0, 2\right) and \left(a, - 6 \right).
Write the equation of the line in the form y = m x + b.
Find the value of a.
Determine whether the following statements about two parallel lines are true or false.
The y-value is changing at the same rate on both lines.
They intersect at one point.
They have the same value of c in y = m x + c.
They have the same value of m in y = m x + c.
They are equidistant from each other.
State whether the given pairs of lines are parallel:
y = - 2 x - 5 and y = - 2 x - 8
y = 7 x + 8 and y = - 5 x + 8
y = - 3 x - 2 and y = - 3 x + 9
y = - 6 x - 5 and y = - 6 x
Find the gradient of the following lines:
A line parallel to a line with gradient - 2.
Any line that is parallel to y = - 7 + 4 x.
State whether the following lines are parallel to y = 7 x + 3.
y = 7 x - 3
y = 6 x + 3
y = 7 x
y = - 7 x + 3
State whether the following lines are parallel to y = - 3 x + 2.
y = 3 x
y = - \dfrac{2 x}{3} + 8
- 3 y - x = 5
y = - 10 - 3 x
For the following graphs:
State the value of the x-intercept.
State the value of the y-intercept.
For each of the following tables of values:
Find the gradient, m.
Find the y-intercept, c.
Write the equation of the line expressing the relationship between x and y.
Complete the table of values.
| x | 0 | 1 | 2 | 3 | 4 | 24 |
|---|---|---|---|---|---|---|
| y | 0 | 4 | 8 | 12 | 16 |
| x | 0 | 1 | 2 | 3 | 4 | 21 |
|---|---|---|---|---|---|---|
| y | 9 | 14 | 19 | 24 | 29 |
| x | 0 | 1 | 2 | 3 | 4 | 25 |
|---|---|---|---|---|---|---|
| y | - 23 | - 21 | - 19 | - 17 | - 15 |
| x | 0 | 1 | 2 | 3 | 4 | 70 |
|---|---|---|---|---|---|---|
| y | 27 | 22 | 17 | 12 | 7 |
For each of the following tables of values:
Find the equation of the line expressing the relationship between x and y.
Complete the table of values.
| x | 1 | 2 | 3 | 4 | 19 |
|---|---|---|---|---|---|
| y | 5 | 10 | 15 | 20 |
| x | 1 | 2 | 3 | 4 | 16 |
|---|---|---|---|---|---|
| y | - 3 | - 6 | - 9 | - 6 |
| x | 1 | 2 | 3 | 4 | 60 |
|---|---|---|---|---|---|
| y | 1 | 4 | 7 | 10 |
| x | 1 | 2 | 3 | 4 | 80 |
|---|---|---|---|---|---|
| y | - 14 | - 22 | - 30 | - 38 |
| x | 1 | 2 | 3 | 4 | -15 |
|---|---|---|---|---|---|
| y | 82 | 74 | 66 | 58 |
Find the equation that corresponds to each of the following tables:
| x | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | 7 | 4 | 1 | - 2 | - 5 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 | 17 |
| x | - 8 | - 7 | - 6 | - 5 | - 4 |
|---|---|---|---|---|---|
| y | - 37 | - 32 | - 27 | - 22 | - 17 |
| x | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| y | -1 | - 3 | - 5 | - 7 | - 9 |
Find the equation of the following lines:
A line that passes through the point A \left( - 5 , - 4 \right) and has a gradient of - 4.
A line that passes through the point A \left( - \dfrac{4}{5} , - 4 \right) and has a gradient of 2.
A line that passes through Point A \left( - 4 , 3\right) and has a gradient of 4.
A line that passes through Point A \left(7, - 6 \right) and has a gradient of - 3.
A line that passes through the point A \left(3, 5\right) and has a gradient of - \dfrac{5}{2}.
A line that passes through the point A \left(4, 3\right) and has a gradient of - 3\dfrac{1}{3}.
A line that passes through the point A \left( - 4 , 3\right) and has a gradient of - 9.
A line that passes through the point A \left( - \dfrac{5}{9} , 7\right) and has a gradient of 7.
A line that passes through the point A \left(8, 1\right), and has a gradient of \dfrac{5}{2}.
A line that passes through the point A \left( - 4 , 5\right) and has a gradient of 3\dfrac{1}{2}.
For each of the following lines:
Find the equation of the line.
Sketch the graph of the line.
A line has gradient 2 and passes through the point \left( - 5 , - 3 \right).
A line has gradient - \dfrac{3}{2} and passes through the point (- 2, 2).
A line has gradient - \dfrac{2}{5} and passes through the point \left( - 10 , 2\right).
A line has gradient - 3 and passes through the point \left(2, - 12 \right).
Consider the line with equation 2 x + y - 8 = 0.
Find the x-intercept of the line.
Hence, find the equation of a line with a gradient of - 4 that passes through the x-intercept of the given line.
Find, in general form, the equation of a line which has a gradient of \dfrac{4}{7} and cuts the x-axis at - 10.
For each of the following lines:
Find the gradient of the line.
Find the equation of the line.
A line passes through the points \left(2, - 7 \right) and \left( - 5 , 6\right).
A line passes through the points \left(3, - 3 \right) and \left(5, - 11 \right).
A line passes through the points A \left( - 6 , 7\right) and B \left( - 8 , - 4 \right).
Identify which of the following equations of straight lines have a gradient of 5 and pass through the point A \left(-1, - 4 \right):
\dfrac{y + 4}{x + 1} = 5
\dfrac{x + 1}{y + 4} = 5
\dfrac{- 4 - y}{-1 - x} = 5
\dfrac{y + 1}{x + 4} = 5
Write down the equations of three lines that pass through the point (1, 3). Explain how your lines are different.
A line has a gradient of \dfrac{3}{10} and passes through the midpoint of A \left( - 6 , - 6 \right) and B \left(8, 8\right).
Find the coordinates of M, the midpoint of AB.
Find the equation of the line in general form.
Consider the lines L_{1}, y = - 4 x + 5, and L_{2}, y = x - 1.
Find the midpoint M of their y-intercepts.
Find the equation of the line that goes through the point M and has gradient \dfrac{1}{3}. Express the equation in general form.
A circle with centre C \left(11, 13\right) has a diameter with end points A \left(5, 14\right) and B \left(p, q\right).
Find the value of p.
Find the value of q.
Find the equation of the line passing through B with gradient \dfrac{9}{2}.