Chapter 3 Review Sections: 3.1, 3.2, 3.3, 3.4
Section 3.1: Solve Linear Systems by Graphing
Key things to remember: Solve each equation for y. Identify y-intercept. Identify slope. Graph both equations on the same graph. Identify if there is exactly one solution = the point where the two lines intersect, solution written as ( x , y ). Check your solution by substituting back into original equation and see if it is true. Identify if there are infinitely many solutions = the two equations are the same line. Identify if there are no solutions = the two equations are parallel. Classify as either: consistent and independent = lines intersect at one point, consistent and dependent = lines coincide, or as inconsistent = lines are parallel.
3.1 Examples Graph the linear system. State your solution. Classify the system as consistent and independent (C&I), consistent and dependent (C&D), or inconsistent. 1. 2x – y = 4 x – 2y = -1 y = 2x – 1 -6x + 3y = -3 y = 3x + 2 y = 3x – 2 Solution is (3,2) and the system is C&I. Infinitely many solutions and the system is C&D. No solution and the system is inconsistent.
Section 3.2: Solve Linear Systems Algebraically
Key things to remember: Substitution Method: Step 1: solve one equation for a variable. Step 2: substitute equation from step 1 into the unused equation and solve for variable. Step 3: substitute the found variable into equation from step 1 and solve. Elimination Method: Step 1: multiply one or both of equations by a number to eliminate a variable. Step 2: combine the two equations and solve for variable. Step 3: substitute back into either of original equations to solve for the other variable. SOLUTION will be a point: (x , y), no solution or infinitely many solutions.
3.2 Examples Solve each system using the substitution or elimination method. 3x + 2y = 1 -2x + y = 4 4x + 3y = -2 x + 5y = -9 3x – 6y = 9 -4x + 7y = -16 Solution is (-1,2). Solution is (1,-2). Solution is (11,4).
Section 3.3: Graph Systems of Linear Inequalities
Key things to remember: Just like when graphing equations, solve for y, identify the y-intercept, identify the slope. Graph the inequalities on the same graph. Check whether the lines are solid or dashed. Then Shade each line either above or below the line (may use different colors). Identify the region that is common to all the graphs.
3.3 Examples 1. x + y ≥ -3 Graph the system of inequalities.
Section 3.4: Solve Systems of Linear Equations in Three Variables
Key things to remember: Step 1: Decide which variable you want to eliminate first from the three equations. Which one looks the easiest? KISS (Keep It So Simple) Step 2: Use two equations to eliminate that variable. Step 3: Use the equation you haven’t used yet and either of the other two equations again to eliminate the same variable. Step 4: Now you have a system of two equations with two of the same variables. Step 5: Use either elimination or substitution to solve for one of the variables. THE HARD PART IS NOW DONE!! Now just substitute back to find other two variables! Step 6: Substitute back into one of the equations from Step 4 to find another variable. Step 7: Substitute the two variables found in Step 5 and Step 6 into any of the 3 original equations and solve for the final variable. SOLUTION will be a point: (x , y, z), no solution or infinitely many solutions.
3.4 Examples Solve the system. 2x + y – z = 9 -x + 6y + 2z = -17
Solution is (3, -1, -4). Solution is (-4, 5, -4).
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