5.1 Equations of Lines Equations of the form ax + by = c are called linear equations in two variables. x y 2 -2 This is the graph of the equation 2x + 3y = 12. (0,4) (6,0) The point (0,4) is the y-intercept. The point (6,0) is the x-intercept.

5.1 Equations of Lines The slope of a line is a number, m, and is defined. m is undefined positive m = 0 If a line goes up from left to right, then it has a positive slope. negative If a line goes down from left to right, then it has a negative slope.

5.1 Equations of Lines y x 2 -2 m is undefined The slope of a line is a number, m, which measures its steepness. m = 2 m = 1 2 m = 0 m = - 1 4

5.1 Equations of Lines y x 1 -1

5.1 Equations of Lines The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula y2 – y1 x2 – x1 m = , (x1 ≠ x2 ). x y (x1, y1) (x2, y2) 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). x2 – x1 y2 – y1 change in y change in x

5.1 Equations of Lines Example: Find the slope of the line passing through the points (2, 3) and (4, 5). y2 – y1 x2 – x1 m = 5 – 3 4 – 2 = = 2 = 1 x y (4, 5) (2, 3) 2 Use the slope formula with x1= 2, y1 = 3, x2 = 4, and y2 = 5. 2

5.1 Equations of Lines State the slope and the y-intercept y = -2x + 3 x + y = 10 2x = 4 y = 5 x + 2y – 5 = 0 y = x2 + 4

5.1 Equations of Lines In 2000, there were 200 Twins fans at McDonell. The number will grow by 20 each year. Write a linear model and sketch the graph.

5.1 Equations of Lines How many fans will there be in 5 years? When will there be 360 fans? 360 = 20t + 200 160 = 20t t = 8 years

5.1 Equations of Lines 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.

5.1 Equations of Lines Example: Graph the line y = 2x – 4. The equation y = 2x – 4 is in the slope-intercept form. So, m = 2 and b = - 4. x y 2. Plot the y-intercept, (0, - 4). 1 = change in y change in x m = 2 3. The slope is 2. (1, -2) 2 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. (0, - 4) 1 The point (1, -2) is also on the line. 5. Draw the line through (0, 4) and (1, -2).

5.2 Equations of Lines Point Slope Write an equation of the line that passes through the point (2,-1) with a slope of 2. A(2,-1)

5.2 Equations of Lines Point Slope Write an equation of the line that passes through the point (-1,-1) with a slope of -3 A(-1,-1)

5.2 Equations of Lines Point Slope Write an equation of the line that passes through the point (1,0) with a slope of 1/2 A(1,0)

5.3 Equations of Lines Two Points Write an equation of the line that passes through the point (2,1) and (3,-3) A(2,1) B(3,-3)

5.3 Equations of Lines Two Points Write an equation of the line that passes through the point (1,4) and (0, 3) A(1,4) B(0,3)

5.3 Equations of Lines Two Points Write an equation of the line that passes through the point (-1,6) and (3, -2) A(-1,6) B(3,-2)

5.3 Equations of Lines Two Points Write an equation of the line that passes through the point (1,-3) and (3, -2) A(-1,-3) B(3,-2)

5.4 Exploring Data: Fitting a Line to Data

5.4 Exploring Data: Fitting a Line to Data Find the equation of the line using 1928 and 1998

5.4 Exploring Data: Fitting a Line to Data Find the equation of the line using 1928 and 1998

5.4 Exploring Data: Fitting a Line to Data

5.4 Exploring Data: Fitting a Line to Data Correlation: The agreement between two sets of data -1 < r < 1 r is called the correlation coefficient

5.5 Standard Form of a Linear Equation

5.5 Standard Form of a Linear Equation Multiply both sides by 2 Subtract 2y from both sides Add 2 to both sides

5.5 Standard Form of a Linear Equation Multiply both sides by 12 Subtract 12y from both sides Add 3 to both sides

5.6 Point-Slope Form of a Line A linear equation written in the form y – y1 = m(x – x1) is in point-slope form. It is used mainly to write the equation of a line. The graph of this equation is a line with slope m passing through the point (x1, y1). The graph of the equation y – 3 = - (x – 4) is a line of slope m = - passing through the point (4, 3). 1 2 x y 4 8 m = - 1 2 (4, 3)

5.6 Point-Slope Form of a Line 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

5.6 Point-Slope Form of a Line Example: Write the slope-intercept form for the equation of the line through the points (4, 3) and (-2, 5). 2 1 5 – 3 -2 – 4 = - 6 3 Calculate the slope. m = y – y1 = m(x – x1) Point-slope form Use m = - and the point (4, 3). y – 3 = - (x – 4) 1 3 Slope-intercept form y = - x + 13 3 1

5.6 Point-Slope Form of a Line Summary of Equations of Lines Slope of a line through two points: Vertical line (undefined slope): x = a Horizontal line (zero slope): y = b Slope-intercept form: y = mx + b Point-slope form: y2 – y1 = m(x2 – x1) Standard form: Ax + By = C