1.1 Lines in the Plane

Objectives:

Write the slope-intercept form of an equation given the slope and a point or two points,

Write the standard form of an equation given the slope and a point, or two points and

Write an equation of a line that is parallel or perpendicular to the graph of a given equation.

Finding Slope-Intercept form

Given two points A(0, b) and C(x, y).

Notice that b is the y intercept of AC. Suppose you need to find the slope of AC.

Now solve the equation for y.

Finding Slope-Intercept form

Multiply each side by x.

Add b to each side

Symmetric property of Equality

You may recognize this as the form of a linear function. When an equation is written in the form y = mx + b, it is in slope-intercept form.

Slope Intercept Form of a Linear Equation

The slope-intercept form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept.

Standard Form of a Linear Equation Remember that the

stand form of the equation of a line is Ax + By = C. Suppose we write this general equation in slope-intercept form.

Subtract Ax from each side.

Divide each side by B.

The slope is and the

y-intercept is , for B ≠ 0.

This can be used to write an equation in standard form when you are given the information you usually use to find the slope-intercept form.

What’s the standard form?

Notes

If you are given the slope and y-intercept of a line, you can find an equation of the line by substituting values of m and b into the slope-intercept form of the equation. Then the equation can be written in standard form. For example, if you know the slope of a line is ⅔ and the y-intercept is 5, an equation of the line is

In standard form, 2x – 3y = -18

Ex. 1: Find the slope-intercept form of the equation of the line that has a slope of ¾ and passes through (8, 2).

You know the slope and the x and y values of one point on the graph. Substitute for m, x and y in the slope-intercept form.

Slope-Intercept form

Substitute values

Multiply

Subtract 6 from both sides

The equation in slope-intercept form is

y = ¾x - 4

Ex. 2: Find the standard form of the equation that passes through (-2, 5) and (3, 1).

First use the two given points to find the slope of the line.

Slope-Intercept form

Substitute values

Multiply

Subtract 6 from both sides

The equation in slope-intercept form is

y = ¾x - 4

Ex. 2: Find the standard form of the equation that passes through (-2, 5) and (3, 1).

First use the two given points to find the slope of the line.

If the slope is -4/5, then –A/B = -(4/5). Thus A = 4 and B = 5

Substitute these values into the standard form. The resulting equation is 4x + 5y = C. Since one of the points on the line is (3, 1), you can substitute these values into the equation to find C.

Ex. 2: Find the standard form of the equation that passes through (-2, 5) and (3, 1).

Standard form of a linear equation.

Substitute values for A and B.

Substitute values for x and y.

Simplify

Addition property of Equality

Ex. 3: Application problem

The atmospheric pressure at sea level is 14.7 pounds per square inch. As divers go deeper into the ocean, the pressure increases. Use the chart to write an equation in slope-intercept form that approximates this relationship. Then find the pressure at 30,000 feet below sea level.

Depth (in feet)

Pressure(lbs/in2)

Sea level (0) 14.7

600 269

1200 536

3000 1338

7200 3208

18,000 8019

How?

Let x represent the ocean dept and y represent the pressure.

Use your calculator and a pair of points to find the slope of the line.

You may want to use another set of points to confirm your slope.

Next?

The slope of the line is approximately 0.445. The y-intercept corresponds to the ocean depth of 0 (at sea level). So the y-intercept is 14.7. An equation that approximates the pressure at certain ocean depths is

y= 0.445x + 14.7 The equation that is derived may differ based on the

set of points used to determine the slope. Use your calculator again to approximate pressure at

30,000.

0.445 x 30,000 + 14.7 = 13364.7

The result is 13,364.7 pounds per square inch.

Ex. 4: Write an equation of the line that passes through (4, 6) and is parallel to the line whose equation is: y = 2/3x + 5.

Parallel lines have the same slope, so the slope of both lines is 2/3. Use the slope-intercept form and the point (4, 6) to find the equation.

Slope-Intercept form

Substitute values for x, y and m

Multiply 2/3 by 4

Common denominator

Subtract 8/3 from 18/3 to get the y-intercept.

The y-intercept is 10/3. An equation of the line is

Ex. 5: Write an equation in standard form for the line that passes through (4, 6) and is perpendicular to the line whose equation is

y=2/3x + 5. The slope of the given line is 2/3. Since

the product of this slope and the slope of a perpendicular line is -1, the slope of a perpendicular line is

-3/2. You can use the slope -3/2 and the point (4,6) to write the equation in standard form.

Standard Form

Substitute values for A, B and x and y

Multiply values

Addition property of equality

Substitute into the standard form.