MAT 105 FALL 2008 Graphs of Linear Functions Chapter 5 Graphs of Linear Functions

Section 5.2 Graphs of Linear Functions MAT 105 FALL 2008 Section 5.2 Graphs of Linear Functions

The steepness of a line is measured by the slope of the line. The slope of a line is the ratio of vertical distance to horizontal distance between two points on a line. m = (x2, y2) (x1, y1)

Examples: Find the slope of the line through the given points. a) (8, 7) and (2, -1) b) (-2.8, 3.1) and (-1.8, 2.6) c) (5, -2) and (-1, -2) d) (7, -4) and (7, 10)

On calculator, look at the following graphs and describe. Y1 = x Y2 = -x Y3 = 2x Y4 = 0.25x SLOPE-INTERCEPT FORM The slope-intercept form of a line: ____________________ Where m is ___________ and (0, b) is the _______________

Graph the equation by hand. 1) 2)

Graph the equation by hand. 3) 4)

Graph the equation by hand. 5) 6)

The only type of linear equation that CANNOT be written in slope-intercept form is that of a ___________________ ___________. It is written in the form of __________________.

Definition: x - intercept The x-intercept of a graph is the points at which the graph intersects the x-axis. The y-coordinate is always ____. The x-intercept is written in the form ( , ).

Definition: y - intercept The y-intercept of a graph is the points at which the graph intersects the y-axis. The x-coordinate is always ____. The y-intercept is written in the form ( , ).

Finding the intercepts of a line algebraically. Given the equation of the line, 1) To find the x-intercept, set y = 0 and solve for x. 2) To find the y-intercept, set x = 0 and solve for y. Always express each intercept as an ordered pair.

Example Algebraically, find the coordinates of the x- and y- intercepts of 5.2x – 2.2y = 78.

Finding the intercepts of a line graphically (using the graphing calculator). 1) Solve the equation for y (do not round off values) 2) Enter the equation into y1 3) To find the x-intercept: use  2: Zero. 4) To find the y-intercept: hit .

Example Use the calculator to find the coordinates of the x- and y- intercepts of 3.6x – 2.1 y = 22.68.

Section 5.3 Solving systems of two linear equations in two unknowns graphically

A system of linear equations consists of two or more equations that share the same variables. Question: How would you describe the solution of a system of two linear equations in two variables? Answer: ______________________________________ ______________________________________________.

Determine whether (2, -5) is a solution of the system

Graphically, a solution of two linear equations in two variables is a point that is on both lines. a.k.a. Point of Intersection! To solve a system of linear equations graphically, 1) Sketch the graph of each line in the same coordinate plane. 2) Find the coordinates of the point of intersection. 3) Check your solution by substituting it back into BOTH of the original equations. Remember, it must satisfy both of the equations.

TWO TYPES OF EQUATIONS WITHIN A SYSTEM: TWO TYPES OF SYSTEMS: 1) Consistent System: A system of equations that has a solution (at least one) 2) Inconsistent System: A system of equations that has NO solution. TWO TYPES OF EQUATIONS WITHIN A SYSTEM: 1) Independent Equations: Equations of a system that have DIFFERENT graphs (For a system of 2 linear equations, this would mean two different lines) 2) Dependent Equations: Equations of a system that have the SAME graph (For a system of 2 linear equations, this would mean the same line)

From these options, three types of systems involving two equations in two variables can be formed.

1. A consistent system of independent equations This system will have exactly one solution (an ordered pair). Graphically, it will look like two lines that intersect at exactly one point. Example:

2. An inconsistent system of independent equations This system will have NO solution. Graphically, it will look like two parallel lines (they never intersect). Notice that these lines have the same slope but a different y-intercept. Example:

3. A consistent system of dependent equations This system will have an infinite number of solutions. Graphically, it will look like a single line. {Think: "Dependent: One leaning on the other“} When the graphs of the two linear equations are the same, we say the lines coincide. Example:

Solve the system graphically (by hand).

Solve the system graphically (using graphing calculator).

Section 5.4 Solving systems of two linear equations in two unknowns algebraically

Solving a system of two linear equations using the SUBSTITUTION METHOD If it is not done already, solve one of the equations for one of its variables (isolate one of the variables). Substitute the resulting expression for that variable into the other equation and solve it. Find the value of the remaining variable by substituting the value found in step 2 into the equation found in step 1. State the solution (an ordered pair). Check the solution in both of the original equations.

Solve using the substitution method:

Solve using the substitution method:

Solve using the substitution method:

Solving a system of two linear equations using the ELIMINATION (ADDITION) method Write both equations in standard form Ax + By = C If necessary, multiply one or both equations by some number(s) in order to make the coefficient of one of the variables opposite. Add the equations to eliminate one of the variables. Solve the equation for the remaining variable. Substitute the value into either of the original equations to find the value of the variable. Check the solution in the original equations.

Solve using the elimination (addition) method:

Solve using the elimination (addition) method:

Solve using the elimination (addition) method:

Solve using the elimination (addition) method:

MAT 105 FALL 2008 Using a matrix to solve a system of linear equations using the graphing calculator (RREF) Refer to handout.

Use a system of equations to solve the problem. A bicycle manufacturer builds racing bikes and mountain bikes, with per unit manufacturing costs as shown in table below. The company has budgeted $31, 800 for labor and $26,150 for materials. How many bicycles of each type can be built? Model Cost of Materials Cost of Labor Racing $110 $120 Mountain $140 $180

Use a system of equations to solve the problem. $9500 is invested into two funds, one paying 8% interest and the other paying 10% interest. The interest earned after one year was $870. How much was invested in each fund?

Systems of THREE linear equations For the following problem: Choose and define three variables to represent the unknown quantities. Using the info from the problem, write three equations involving the variables. Solve the system of equations using your calculator to find the reduced row-echelon form of the augmented matrix. State the solution to the problem in words. Include the appropriate units. Do NOT include the variables.

Use a system of equations to solve the problem. McDonald’s recently sold small soft drinks for $0.87, medium soft drinks for $1.08, and large soft drinks for $1.54. During a lunchtime rush, Chris sold 40 soft drinks for a total of $43.40. The number of small and large drinks, combined, was 10 fewer than the number of medium drinks. How many drinks of each size were sold?