Unit C Solving Systems of Equations by Graphing

ACT WARM-UP There are 30 antique cars in a parade. Six of the cars are red, 14 are black, 5 are blue, and 5 are white. If a circle graph is used to represent this information, what percent of the graph would accurately represent the number of red cars? A) 5 B) 6 C) 16.7 D) 20 E) 24 6 out of 30 are red. This is 1/5 of the total, therefore 1/5 = .2 or 20%. The answer is D) 20.

Objectives Solve systems of linear equations by graphing. Classify systems of equations, and determine the number of solutions.

How do you solve a system of equations by graphing? Essential Question: How do you solve a system of equations by graphing?

A system of equations is two or more equations containing two or more variables. A linear system is a system of equations containing only linear equations. We will examine other systems later. The solution of a system of equations is the set of all points that satisfy each equation. Use substitution to determine if an ordered pair is an element of the solution set for a system of equations. On the graph of the system of two linear equations, the solution is the set of points where the lines intersect on the same coordinate plane.

Solve the system of equations by graphing. Write each equation in slope-intercept form or standard form using x- and y- intercepts and/or slope and then graph. The graphs appear to intersect at (4, 2). Example 1-1a

Check Substitute the coordinates into each equation. Original equations Replace x with 4 and y with 2. Simplify. Answer: The solution of the system is (4, 2). Example 1-1a

Solve the system of equations by graphing. Answer: (4, 1) Example 1-1b

Systems of equations are used in businesses to determine the break-even point. The break-even point is the point at which the income equals the cost. If a business is operating at the break-even point, it is neither making nor losing money.

Fund-raising A service club is selling copies of their holiday cookbook to raise funds for a project. The printer’s set-up charge is $200, and each book costs $2 to print. The cookbooks will sell for $6 each. How many cookbooks must the members sell before they make a profit? Let Cost of books is cost per book plus set-up charge. y = 2x + 200 Example 1-2a

Income from books is price per book times number of books. y = 6 x Answer: The graphs intersect at (50, 300). This is the break-even point. If the group sells less than 50 books, they will lose money. If the group sells more than 50 books, they will make a profit. Example 1-2a

Classify Systems of Equations Graphs of systems of linear equations may be intersecting lines, parallel lines, or the same line. A system of equations is consistent if it has at least one solution and inconsistent if it has no solutions. A consistent system is independent if it has exactly one solution or dependent if it has an infinite number of solutions.

Systems of Equations A system is consistent if it has one or more solutions.

Systems of Equations A system is inconsistent if it has no solutions.

Systems of Equations A consistent system is independent if it has exactly one solution.

Systems of Equations A consistent system is dependent if it has an infinite number of solutions. Two equations that represent the same line are said to coincide.

Systems of Equations

Since the equations are equivalent, their graphs are the same line. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Since the equations are equivalent, their graphs are the same line. Any ordered pair representing a point on that line will satisfy both equations. So, there are infinitely many solutions. This system is consistent and dependent. Example 1-4a

Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Answer: inconsistent Example 1-5b

How do you solve a system of equations by graphing? Essential Question: How do you solve a system of equations by graphing? Graph both equations on the same coordinate plane. The point(s) of intersection are the solutions of the system.

Math Fact The term pencil can be used to describe the set of all lines that pass through a given point. A pencil may be composed of many consistent, independent linear systems.