Algebra-2 Section 3-2A Solving Systems of Linear Equations Algebraically Using Substitution

Algebra-2Algebra-2

Section 3-2ASection 3-2A

Solving Systems of Linear Solving Systems of Linear Equations Algebraically Using Equations Algebraically Using Substitution.Substitution.

Quiz 3-1Quiz 3-11. Solve the system by graphing: 1. Solve the system by graphing:

1xy 12

1 xy

2.2. Check the following ordered pair to see if it is a Check the following ordered pair to see if it is a solution to the following system of equations solution to the following system of equations (2, 1) (2, 1)

732 yx

74 xy

3-2: Solve Linear Systems 3-2: Solve Linear Systems AlgebraicallyAlgebraically

ReviewReview

22 yx

Solve for ‘x’Solve for ‘x’

- 2y - 2y - 2y - 2y

yx 22

Your turn:Your turn:

11. Solve for ‘x’. Solve for ‘x’

22. Solve for ‘y’. Solve for ‘y’

1553 yx

8412 yx

If I gave you a choice:If I gave you a choice: Which variable would be easier to solve for (‘x’ or ‘y’)?Which variable would be easier to solve for (‘x’ or ‘y’)?

155 yx

845 yx

Which variable would be easier to solve for (‘x’ or ‘y’)?Which variable would be easier to solve for (‘x’ or ‘y’)?

Your turn:Your turn:What is the What is the easiesteasiest variable to solve for in each of variable to solve for in each of the following equations (‘x’ or ‘y’)?the following equations (‘x’ or ‘y’)?

3. 3.

4.4.

5. 5. 1553 yx

87 yx

83 yx

VocabularyVocabularySystems of linear equationsSystems of linear equations: Two or more equations : Two or more equations

(of lines) that each have the same two variables. (of lines) that each have the same two variables.

Ax + By = C (equation 1)Ax + By = C (equation 1)Dx + Ey = F (equation 2)Dx + Ey = F (equation 2)

3x + y = 73x + y = 75x - 2y = -35x - 2y = -3

22 xy

64 xy

xy 46 87 xy

Your turn:Your turn:Solve for ‘y’ for the following equation if: x = 3Solve for ‘y’ for the following equation if: x = 3

6.6.

7. 7. 862 yx

83 yx

In these problems we In these problems we substitutedsubstituted a variable with a a variable with a number in order to solve for the other variable. number in order to solve for the other variable.

VocabularyVocabularySubstitution Method for solving systems of equationsSubstitution Method for solving systems of equations:: (1)(1) Solve one of the equations for one of the variables. Solve one of the equations for one of the variables.

(2) Replace or “Substitute” the variable in (2) Replace or “Substitute” the variable in the second equation with the equivalent expression forthe second equation with the equivalent expression for that variable that you found in step (1)that variable that you found in step (1)

(3) Solve this single variable equation. (3) Solve this single variable equation.

(4) Plug the numerical value of this variable into either of the (4) Plug the numerical value of this variable into either of the original equations to solve for the other variable.original equations to solve for the other variable.

Substituting a Variable with an Substituting a Variable with an expression.expression.

643 yx

3(-2 – 2y) (-2 – 2y) + 4y = 6

22 yx (1)(1) Solve one of the equations Solve one of the equations for one of the variables. for one of the variables.

yx 22 (2) “Substitute” the variable (2) “Substitute” the variable in the second equation with in the second equation with the equivalent expression the equivalent expression for that variable that you for that variable that you found in step (1)found in step (1)


x = -2 – 2yx = -2 – 2y

643 yx

3(-2 – 2y) (-2 – 2y) + 4y = 6

3( -2) – (3)(2y) + 4y = 63( -2) – (3)(2y) + 4y = 6

Now what?Now what?

-6 – 6y + 4y = 6-6 – 6y + 4y = 6

-6 – 2y = 6-6 – 2y = 6-2y = 12-2y = 12

y = -6 y = -6


x = -2 – 2yx = -2 – 2y

y = -6 y = -6 x = ?x = ?

643 yx

““Substitution Substitution step”step”

Substitute -6 into one (or the other)Substitute -6 into one (or the other)of the original equations.of the original equations.

Which equation is Which equation is easier to solve for ‘x’?easier to solve for ‘x’?

x = -2 – 2(x = -2 – 2(-6)-6) x = 10x = 10

Solution: (10, -6)Solution: (10, -6)

3 x + 2y = 63 x + 2y = 6

124 yx

Another Example:Another Example:

x = 4y – 12x = 4y – 12

3(3( ) + 2y = 6) + 2y = 6

x = ?x = ?

x – 4y = -12 x – 4y = -12 3 x + 2y = 63 x + 2y = 6

1.1. Solve one of the equations for ‘x’ (or ‘y’ whichever is easier).Solve one of the equations for ‘x’ (or ‘y’ whichever is easier).

2. Substitute ‘x’ in the other equation with the expression that equals ‘x’.2. Substitute ‘x’ in the other equation with the expression that equals ‘x’.

124 y

3. Solve for ‘y’3. Solve for ‘y’

3(4y – 12) + 2y = 63(4y – 12) + 2y = 6

12y – 36 + 2y = 612y – 36 + 2y = 6

14y – 36 = 614y – 36 = 6

14y = 4214y = 42 y = 3y = 3

Substitution step:Substitution step:

y = 3y = 3Substitute ‘y’ with 3 in any of the Substitute ‘y’ with 3 in any of the original (or equivalent) equations. original (or equivalent) equations.

x – 4y = -12 x – 4y = -12 3 x + 2y = 63 x + 2y = 6

x = 4y – 12x = 4y – 12

x = 4x = 4(3) (3) – 12– 12

Now, solve for ‘x’.Now, solve for ‘x’.

x = 0x = 0

The solution to the system is:The solution to the system is:

(0, 3)(0, 3)

10. 10. Substitute the expression that is equivalent to this variableSubstitute the expression that is equivalent to this variable into the other equation.into the other equation.

Your turn:Your turn:8. 8. Identify the equation that is the easiest to solve for one of Identify the equation that is the easiest to solve for one of the variables. the variables. 24 yx

75)24(2 yy

752 yx24 yx

9. 9. Solve this equation for the easiest variable. Solve this equation for the easiest variable.

24 yx

Your turn:Your turn:75)24(2 yy

752 yx24 yx

11. 11. Solve this equation for the one variable.Solve this equation for the one variable.

12. 12. Substitute the numerical value of this variable into Substitute the numerical value of this variable into the equation found in problem #5 above. the equation found in problem #5 above.

13. 13. Solve for ‘x’.Solve for ‘x’.

1y

2)1(4 x

6x

14. 14. Write the solution.Write the solution. )1 ,6(

Your turn:Your turn:15. 15. Solve using “substitution”Solve using “substitution”

1832 yx13 yx

13 xy

18)13(32 xx

18392 xx

1837 x

217 x3x

1)3(3 y

19 y8y

Solution: (-3, 8)Solution: (-3, 8)

Your turn:Your turn:16. 16. Solve using “substitution”Solve using “substitution”

12 yx42 yx

12 yx

4)12(2 yy424 yy

423 y63 y

2y

1)2(2 x

14 x3x

Solution: (3, 2)Solution: (3, 2)

Your turn:Your turn:32 yx23 xy

1717. . Solve the system of equationsSolve the system of equations

)1 ,1(

2)1(3 y23y

1y

23 xy3)23(2 xx

3232 xx

325 x

55 x

1x

Your turn:Your turn:84 xy

42 xy1818. . Solve the system of equationsSolve the system of equations

)0 ,2(

4)2(2 y0y

8442 xx

846 x

126 x

2x

Categories of Solutions:Categories of Solutions:

Ways 2 lines can be graphed:Ways 2 lines can be graphed:

Cross Cross one solution one solution

Parallel Parallel no solutionsno solutions

Same line Same line infinite infinite number of number of solutionssolutions

3 3 ClassesClasses of solutions: of solutions:1. The lines intersect: 1. The lines intersect: 1 solution1 solution

2. The lines do not intersect: 2. The lines do not intersect:

3. The lines are 3. The lines are coexistentcoexistent: :

0 solutions0 solutions

Infinite #Infinite #of solutionsof solutions

How does the substitution method tell you there are How does the substitution method tell you there are zerozero or an or an infinite infinite number of solutions?number of solutions?

Example:Example: 526 yx 73 yx

Which of the two equations is it easiest to solve Which of the two equations is it easiest to solve for one of the variables? for one of the variables?

73 yx

Solve for ‘y’ in that equation:Solve for ‘y’ in that equation: 73 xy

Substitution stepSubstitution step 526 yx

5)73(26 xx

51466 xx

514 WHAT???!!!WHAT???!!!

The variable The variable disappearsdisappears and and the statement is the statement is falsefalse. . no solution (lines are parallel)no solution (lines are parallel)

Solve for ‘x’Solve for ‘x’

How do you know?How do you know? (1, 0, or infinite #) (1, 0, or infinite #)

Using the substitution method, if the variable Using the substitution method, if the variable “disappears” and the resulting equation is either:“disappears” and the resulting equation is either:

b. b. truetrue:: (3 = 3 or 0 = 0)(3 = 3 or 0 = 0) Infinite # of solutionsInfinite # of solutions

a. a. falsefalse:: (-2 = 3 or 10 = 0)(-2 = 3 or 10 = 0) No solutionNo solution

BUT: it’s easier to check the original equations to see ifBUT: it’s easier to check the original equations to see if (1) they are parallel (no solution) or (2) the same line(1) they are parallel (no solution) or (2) the same line (infinitely many solutions).(infinitely many solutions).

Your turn:Your turn:1919. Solve:. Solve:

2x + y = -22x + y = -25x + 3y = -85x + 3y = -8

How do you know how many How do you know how many solutions there are? (1, 0, or solutions there are? (1, 0, or infinite #)infinite #)

13 xy12 xy

Not same line, not parallel Not same line, not parallel one solution. one solution.

32 xy42 xy

parallel parallel no solutions no solutions

222 yx

1 yx

11stst equation is a multiple of the 2 equation is a multiple of the 2ndnd equation equation same line same line

infinite # of solutions.infinite # of solutions.

Which Which Category ?Category ?




62 xy

24 xy





42 xy

72 xy





632 yx

1264 yx


20. 20. Which category ?Which category ?




824 yx

63 yx





1523 yx

2

15

2

3 xy21. 21. Which category?Which category?





1y

53 yx22. 22. Which category?Which category?

Vocabulary:Vocabulary:SolutionSolution: (single variable equation). The number you can : (single variable equation). The number you can

substitute into the equation to make it a true statement.substitute into the equation to make it a true statement.

242 x 62 x3xCheckCheck 24)3(2

246 22

x = 2 is a solution of the equation: 2x – 4 = 2 because if x = 2 is a solution of the equation: 2x – 4 = 2 because if you replace ‘x’ in the equation with 2, left side equals right you replace ‘x’ in the equation with 2, left side equals right side (the equation is a true statement).side (the equation is a true statement).

Vocabulary:Vocabulary:SolutionSolution: (two variable equation). The ordered pair (values : (two variable equation). The ordered pair (values

for ‘x’ and ‘y’) that you can substitute into the equation for ‘x’ and ‘y’) that you can substitute into the equation to make it a true statement.to make it a true statement.

42 xy

4)0(2 y

2x

Plug in y = 0 for Plug in y = 0 for the x-interceptthe x-intercept

4y

420 xx24

(2, -4) is a solution of the equation: y = 2x - 4 because if (2, -4) is a solution of the equation: y = 2x - 4 because if you replace ‘x’ with 2 and ‘y’ with -4, the left side of the you replace ‘x’ with 2 and ‘y’ with -4, the left side of the equation equals the right side of the equation (the equation equation equals the right side of the equation (the equation is a true statement).is a true statement).

There are an infinite number of pairs.There are an infinite number of pairs.

Plug in x = 0 for Plug in x = 0 for the y-interceptthe y-intercept

Vocabulary:Vocabulary:Solution to system of equationsSolution to system of equations: The : The ordered pairordered pair (‘x’ and (‘x’ and

‘y’ values) that you can substitute into both equations to ‘y’ values) that you can substitute into both equations to bothboth equations into true statements. equations into true statements.

(6, -1) is a solution to the system of equations: x + 4y = 2, (6, -1) is a solution to the system of equations: x + 4y = 2, and 2x + 5y = 7 because if you replace ‘x = 6’ and ‘y = -1’ and 2x + 5y = 7 because if you replace ‘x = 6’ and ‘y = -1’ into into bothboth equations, it makes both equations true equations, it makes both equations true statements. statements.

24 yx 752 yx

2)1(4)6(

246 22

752 yx7)1(5)6(2

7512 77

Your turn:Your turn:23. 23. Is the ordered pair (3, -1) a solution of the following Is the ordered pair (3, -1) a solution of the following

system of equations? system of equations?

822 yx

332 yx

24. 24. Is the ordered pair (2, 1) a solution of the following Is the ordered pair (2, 1) a solution of the following system of equations? system of equations? 42 yx

232 yx

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Algebra-2 Section 3-2A Solving Systems of Linear Equations Algebraically Using Substitution