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Systems of Linear Equations Math 0099 Section 8.1-8.3 Created and Presented by Laura Ralston
What is a system of linear equations? • It is two or more linear equations with the same variables considered at the same time. • Number of variables equals number of linear equations in the system • Examples: x + y = 4 x + y = 2 x+ y = -10 3x + 4y = 7
What is the solution set to a system? • The solution set to the system of linear equations is ALL ordered pairs that are solutions to both equations, that is, makes both equations TRUE at the same time. • To decide whether an ordered pair is a solution to a system, substitute the values for x and y in both equations. If the results for both equations are true then the ordered pair is a solution to the system.
Example: Determine if the given point is a solution to the system. x + y = 2 3x + 4y = 7 (1, 1) (4, -2)
Questions to Answer • How do we find the solution, if there is one? • Will there always be a solution to a system of linear equations? • Can there be more than one solution?
Methods for Solving a System of Equations • Graphing - Section 8.1 • Substitution - Section 8.2 • Addition (Elimination) Section 8.3
GRAPHING Procedure • Graph the first equation in the coordinate plane • Graph the second equation on the same coordinate plane • Record the coordinates of the point of intersection of the two graphs. This ordered pair is the solution to the system • Check solution in both equations.
Three possibilities for solutions for a system • NO SOLUTION • Graphically, the lines would be parallel. • Solving for x will result in a false statement with no variable remaining • INCONSISTENT
ONE SOLUTION • Graphically, the lines will intersect ONCE. Solution will be an ordered pair • Solving for x will result in a numerical value CONSISTENT
INFINITE SOLUTIONS • Graphically, the lines coincide (same line) • Solving for x results in a true statement with no variable remaining • DEPENDENT
Examples • x + 2y = 8 • 2x – y = 1 • y = 2x + 5 • 4x – 2y = -10
Assignment Page 595 #1-7 odd, 13-39 odd
SUBSTITUTION • Objective is to eliminate one of the variables so that a new equation is formed with just one variable • Most useful when one of the equation is solved for one variable already OR if one of the variables has a coefficient of 1; otherwise, you get Fractions !!! Fractions !!! Fractions !!! • Provides exact answers rather than estimations
Substitution Steps • Solve one of the given equations for either x or y, whichever is easier. • Substitute the result from step 1 into the other given equation • Solve for the remaining variable • Substitute (“back substitute”) this solution into one of the ORIGINAL given equations
Substitution steps continued ... • Solve for the variable. Write final solution as an ordered pair (x, y) • Check answer in both given equations. True statements indicate correct answers.
Examples • x + y =3 y = -3 =2x • y = 2x 4x – 2y = 6 • y = 4 – 3x • Y=-3x + 6
Assignment Page 603 #1-41 odd
ADDITION (ELIMINATION) • The idea is to eliminate one of the variables from the system of linear equations. • To do this, one of the variables must have coefficients that are opposites. • Provides exact answers rather than estimated ones
Addition (Elimination) Steps • Write each equation in standard form (align like terms) • If needed, multiply one or both equations by appropriate number(s) so that the coefficients on either x or y are opposites. • Add the equations from step 2 together by combining like terms. This should result in an equation with one variable. • Solve the equation from step 3.
Addition steps continued….. • Back Substitute the solution from step 4 into either of the ORIGINAL given equation • Solve for the other variable. Write final answer in an ordered pair (x, y) • Check your answer in each original given equation. True statements result in correct answers.
Examples • 2x + 2y = 4 • x – y = -3 • y = 3x + 15 • 6x – 2y = -30 • 2x – 5y = 6 • 4x – 10y = -2
Assignment Page 611 #1-41 odd
COMPASS Practice Questions What is the solution of the system of equations below? A. (3a, 2a) B. (-3a, 2a) 3x + 4y = a C. (15a, 11a) 2x – 4y = 14a D. (15a, -11a) E. (3a, -2a)
What are the (x, y) coordinates of the point of intersection of the lines determined by the equations 2x – 3y = 4 and y = x? A. (4, 4) B. (–4, –4) C. (–4, 4) D. (4, –4) E. (2, 0)