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Text Box: Math 1314
College Algebra

LO-6
Chapters 2.3

 

 

 

 

 

 

 

 

 

 

 

 

 

                                    


Skills to Master

·       Know and use the point-slope equation of a line to write linear equations given either the point and slope of the line or two points on the line

·       Know and use the slope-intercept equation of a line to write equations of a line given either the slope and y-intercept of the line or the slope and any point on the line

·       Graph equations given in slope-intercept form

·       Determine the slope and y-intercept for a line given in standard (general) form


·        

Point-Slope form of a Linear Equation

In the previous section, we used the slope-intercept form of a linear equation.  That form is , where m is the slope of the line and b is the y-intercept.  A slightly different form is called the point-slope form.  It is , where as before m is the slope of the line and  and  are the coordinates of a point on the line.  The y and the x are treated as variables.  That is, whereas you would substitute actual number values for  and , you would leave y and x as variables when using this equations.

This form is derived from the slope formula.  If we say that x and y are any points on some line with slope m, and  is some particular point on the line, then we have:

We can cancel the numerator by multiplying both sides:

Cancelling on the right gives:

, which is the same as

This form allows us to write the equation of a line when the slope is known and when at least one point is known.

Example

What is the equation of the line with slope m = 3 and passing through the point (4, -1)?

 

What is the equation of the line with slope  and passing through the point (-3, 2)?

 

This equation can also be used to write the equation of a line when given only two points:

Example

Write the equation of the line passing through the points (1, -6) and (-2, -9)

 

Write the equation of the line passing through the points (4, -2) and (-4, 4)

Slope-Intercept form of a Linear Equation

Yesterday you learned a bit about the slope-intercept form: .  Today you learned that, in this form, the value of m is the slope of the line.  Notice that when the value of x is zero, then .  That is, the point associated with  is .  But recall from yesterday that this is the special point called the y-intercept.  Thus, b is the y-intercept of the line.

IMPORTANT: the slope-intercept form of a linear equation gives the values of the slope (m) and the y-intercept (b) directly. 

Example

Write the equation of the line with slope  and y-intercept

 

Write the equation of the line with slope  and y-intercept

 

We can also use this form to find the equation of a line when given the slope and a point on the line, like one of the problems above.  The difference is that using this form produces only the value of b.

Example

Find the equation of the line with slope m = 2 and passing through (1, 9)

 

Find the equation of the line with slope m = -1 and passing through (4, -4)

 

Standard Form of the Equation of a Line

In the previous section you learned that another form of a linear equation is , where A, B, and C are integers.  Unlike the slope-intercept form, it is not possible to directly read the values of m and b (the slope and the y-intercept) from the standard form.  However, we can determine from the equation how to find these values by solving the equation for y:

Since we want to isolate the y, we must first replace the value of Ax with zero.  We do this by subtracting Ax from both sides of the equation.  This gives:

Now we want to replace the value of B with a 1.  We do this by dividing.  Remember to divide all terms on both sides by B!  This gives:

Compare this to .  Which part is like m?  Which part is like b?

 

We see, then, that the slope is  and the y-intercept is .

Example

What is the slope of the line and the y-intercept for the equation ?

 

What is the slope of the line and the y-intercept for the equation ?

 

To complete this section, let’s look at two types of problems that require you to use several of the concepts about linear equations that you’ve studied so far:

Example

What is the equation of the line that passes through the point (4, 3) and that is parallel to the line ?

 

What is the equation of the line passing through the point (-1, -1) and perpendicular to the line ?

 

Try these:

What is the equation of the line passing through the point (0, -4) and parallel to the line ?

What is the equation of the line passing through the point (2, 4) and perpendicular to the line ?