Substitution and Elimination

1. Substitution and Elimination Lesson 3.6

2. A solution to a system of equations in two variables is a pair of values that satisfies both equations and represents the intersection of their graphs. • In Lesson 3.6, you reviewed solving a system of equations using substitution, when both equations are in intercept form. • Suppose you want to solve a system and one or both of the equations are not in intercept form. You can rearrange them into intercept form, but sometimes there’s an easier method. • If one equation is in intercept form, you can still use substitution.

3. Solve this system for x and y -10x-5y=-30 Original form of the second equation. -10x-5(15+8x)=-30 Substitute the right side of the first equation for y. -10x-75-40x = -30 Distribute -5. -50x = 45 Add 75 to both sides and combine like terms. x = -0.9 Now that you know the value of x, you can substitute it into either equation to find the value of y. y = 15+8(-0.9) Substitute 0.9 for x in the first equation. y = 7.8 Multiply and combine like terms.

4. The substitution method relies on the substitution property, which says • that if a= b, then a may be replaced by b in an algebraic expression. • Substitution is a powerful mathematical tool that allows you to rewrite expressions and equations in forms that are easier to use and solve. Notice that substituting an expression for y, as you did in Example A, eliminates y from the equation, allowing you to solve a single equation for a single variable, x.

5. A third method for solving a system of equations is the elimination method. • Theelimination method uses the addition property of equality, which says that • if a=b and c=d, then a+c=b+d.In other words, if you add equal quantities to both sides of an equation, the resulting equation is still true. • If necessary, you can also use the multiplication property of equality, which says that • if a=b, thenac=bc,or if you multiply both sides of an equation by equal quantities, then the resulting equation is still true.

6. Example A • Solve this system for x and y. You can substitute the coordinates back into both equations to check that the point is a solution for both.

7. Example A • Solve this system for x and y. You can substitute the coordinates back into both equations to check that the point is a solution for both.

8. What’s Your System? • In this investigation you will discover different classifications of systems and their properties. You can divide up the work among group members, but make sure each problem is solved by one person and checked by another. • Use the method of elimination to solve each system. (Don’t be surprised if it doesn’t always work.)

9. Graph each system. • A system that has a solution (a point or points of intersection) is called consistent. Which of the six systems are consistent? • A system that has no solution (no point of intersection) is called inconsistent. Which of the systems are inconsistent? • A system that has infinitely many solutions is called dependent. For linear systems, this means the equations are equivalent (though they may not look identical). • A system that has a single solution is called independent. Which ofthe systems are dependent? Independent?

10. Consistent a b d e

11. Inconsistent c a

12. Dependent d b Independent a e

13. Your graphs helped you classify each system as inconsistent or consistent and as dependent or independent. Now look at your solutions. • Make a conjecture about how the results of the elimination method can be used to classify a system of equations.