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## Determining Linear Equations (or Functions) from Information about the Line

If we are given a straight-line graph, we can find the slope m and intercept b by direct measurement, and hence write down the equation. More usually, we are given some information about the line which allows us to calculate m and b and hence write the equation for the line. A linear function can be determined in the following cases:

• Case 1: The slope and a point on the line are known.
• Case 2: Two points on the line are given.

### Example 6A.

Case 1: We are given the slope and a point and asked to find the equation of the line:

 Suppose a line has the slope m = and passes through the point ( , ) .   Click on the step buttons to show how to find the linear function.   First substitute the slope into the equation of a line: y = x + b Next, substitute the point into the equation and solve for b: = ( ) + b Solving for b we find:          b = y = x +

### Example 6B.

Here we look at a special case of the above example where we are given the slope and the x-intercept. As the x-intercept defines a point, we again have the slope of a line and a point on the line and so we can find the linear function.

 Let the slope of a line be   m = and x-intercept be   x = Click on the step buttons to show how to find the linear function.   First substitute the slope into the equation of a line: y = x + b Next substitute the point ( , 0) into the equation: = ( ) + b                   Hence b = y = x +

Note: If we are given the slope and the y-intercept, finding the equation of the line y = mx + b is straighforward as we have both m and b and can substitute these directly into the equation.

### Exercise 6.

In the following exercise you are given the slope and one of the following: a general point on the line, the x-intercept or the y-intercept. Use the information given to determine the linear function.

Find the linear equation given any two of the following facts:

 Slope Point x-intercept y-intercept Function ( , ) ( , ) ( , ) y = x +

### Example 7.

Case 2: We are given two points and asked to find the equation of the line that goes through these two points:

If the line passes through ( , ) and ( , ),  then the slope of the line is

 - m = = to one decimal place. -

Click on the step buttons to show how to find the linear function.

First substitute the slope into the equation of a line:

y = x + b

Next, substitute one of the two points into the equation and solve for b:

= ( ) + b

Hence b = to one decimal place.

y = x +

### Exercise 7.

Find the linear equation given either two points or an x- and y- intercept:

 Point Point x-intercept y-intercept Function ( , ) ( , ) ( , ) ( , ) y = x +

Give your answers to 2 decimal places.

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