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Systems of Linear Equations Math 0099 Section 8.1-8.3 Section 8.1-8.3 Created and Presented by Laura Ralston
What is a system of linear equations? n It is two or more linear equations with the same variables considered at the same time. n Number of variables equals number of linear equations in the system n Examples: x + y = 4 x + y = 2 x+ y = -103x + 4y = 7
What is the solution set to a system? n 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. n 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 n How do we find the solution, if there is one? n Will there always be a solution to a system of linear equations? n 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 Section 8.3
GRAPHING Procedure 1. Graph the first equation in the coordinate plane 2. Graph the second equation on the same coordinate plane 3. Record the coordinates of the point of intersection of the two graphs. This ordered pair is the solution to the system 4. Check solution in both equations.
Three possibilities for solutions for a system n NO SOLUTION –Graphically, the lines would be parallel. –Solving for x will result in a false statement with no variable remaining –INCONSISTENT
n ONE SOLUTION –Graphically, the lines will intersect ONCE. Solution will be an ordered pair –Solving for x will result in a numerical value CONSISTENT
n INFINITE SOLUTIONS –Graphically, the lines coincide (same line) –Solving for x results in a true statement with no variable remaining –DEPENDENT
Examples n x + 2y = 8 n 2x – y = 1 n y = 2x + 5 n 4x – 2y = -10
SUBSTITUTION n Objective is to eliminate one of the variables so that a new equation is formed with just one variable n 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 !!! n Provides exact answers rather than estimations
Substitution Steps 1 Solve one of the given equations for either x or y, whichever is easier. 2 Substitute the result from step 1 into the other given equation 3 Solve for the remaining variable 4 Substitute (“back substitute”) this solution into one of the ORIGINAL given equations
Substitution steps continued... 5 Solve for the variable. Write final solution as an ordered pair (x, y) 6 Check answer in both given equations. True statements indicate correct answers.
Examples n x + y =3y = -3 =2x n y = 2x 4x – 2y = 6 n y = 4 – 3x n Y=-3x + 6
ADDITION (ELIMINATION) n The idea is to eliminate one of the variables from the system of linear equations. n To do this, one of the variables must have coefficients that are opposites. n Provides exact answers rather than estimated ones
Addition (Elimination) Steps 1 Write each equation in standard form (align like terms) 2 If needed, multiply one or both equations by appropriate number(s) so that the coefficients on either x or y are opposites. 3 Add the equations from step 2 together by combining like terms. This should result in an equation with one variable. 4 Solve the equation from step 3.
Addition steps continued….. 5 Back Substitute the solution from step 4 into either of the ORIGINAL given equation 6 Solve for the other variable. Write final answer in an ordered pair (x, y) 7 Check your answer in each original given equation. True statements result in correct answers.
Examples n 2x + 2y = 4 n x – y = -3 n y = 3x + 15 n 6x – 2y = -30 n 2x – 5y = 6 n 4x – 10y = -2
COMPASS Practice Questions What is the solution of the system of equations below? A. (3a, 2a) A. (3a, 2a) B. (-3a, 2a) 3x + 4y = a B. (-3a, 2a) 3x + 4y = a C. (15a, 11a) 2x – 4y = 14a C. (15a, 11a) 2x – 4y = 14a D. (15a, -11a) D. (15a, -11a) E. (3a, -2a) E. (3a, -2a) n
n 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)