Pc Linear Equations

Linear Equations in Two Variables

Equations of the form ax + by = c are called linear equations in two variables. y

This is the graph of the equation 2x + 3y = 12.

(0,4) (6,0) -2

The point (0,4) is the y-intercept. The point (6,0) is the x-intercept.

2

x

The slope of a line is a number, m, which measures its steepness. y

m is undefined

m=2 m=

1 2

m=0 x -2

2

m=-

1 4

The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula y2 – y1 , (x1 ≠ x2 ). m= x2 – x1 The slope is the change in y divided by the change in x as we move along the line from (x1, y1) to (x2, y2).

y

(x2, y2)

(x1, y1)

y2 – y1 change in y x2 – x1 change in x

x

Example: Find the slope of the line passing through the points (2, 4) and (4, 5). Use the slope formula with x1= 2, y1 = 4, x2 = 4, and y2 = 5.

y2 – y1 5–4 m= = x2 – x1 4–2 y

2 = =1 2

(4, 5) 2

(2, 3) 2

x

A linear equation written in the form y = mx + b is in slope-intercept form. The slope is m and the y-intercept is (0, b). To graph an equation in slope-intercept form: 1. Write the equation in the form y = mx + b. Identify m and b. 2. Plot the y-intercept (0, b). 3. Starting at the y-intercept, find another point on the line using the slope. 4. Draw the line through (0, b) and the point located using the slope.

Example: Graph the line y = 2x – 4. • The equation y = 2x – 4 is in the slope-intercept form. So, m = 2 and b = - 4. y 2. Plot the y-intercept, (0, - 4).

x

change in y 2 = 3. The slope is 2. m = 1 change in x 4. Start at the point (0, 4). Count 1 unit to the right and 2 units up to locate a second point on the line. The point (1, -2) is also on the line. 5. Draw the line through (0, 4) and (1, -2).

(1, -2) 2

(0, - 4) 1

A linear equation written in the form y – y1 = m(x – x1) is in point-slope form. The graph of this equation is a line with slope m passing through the point (x1, y1). y

Example: The graph of the equation

8

1 y – 3 = - (x – 4) is a line 2 1

1 m=2

4

(4, 3)

of slope m = - passing 2

through the point (4, 3).

x 4

8

Example: Write the slope-intercept form for the equation of the line through the point (-2, 5) with a slope of 3. Use the point-slope form, y – y1 = m(x – x1), with m = 3 and (x1, y1) = (-2, 5). y – y1 = m(x – x1)

Point-slope form

y – y1 = 3(x – x1)

Let m = 3.

y – 5 = 3(x – (-2))

Let (x1, y1) = (-2, 5).

y – 5 = 3(x + 2)

Simplify.

y = 3x + 11

Slope-intercept form

Example: Write the slope-intercept form for the equation of the line through the points (4, 3) and (-2, 5). 5–3 =- 2 =- 1 m= -2 – 4 6 3 y – y1 = m(x – x1)

1 (x – 4) 3 y = - 1 x + 13 3 3 y–3=-

Calculate the slope. Point-slope form Use m = -

1 and the point (4, 3). 3

Slope-intercept form