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11-6 Systems of Equations
Course 3
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Warm UpSolve for the indicated variable.
1. P = R – C for R
2. V = Ah for A
3. R = for C
R = P + C
Rt + S = C
Course 3
11-6 Systems of Equations
13C – S
t
= A3Vh
Problem of the Day
At an audio store, stereos have 2 speakers and home-theater systems have 5 speakers. There are 30 sound systems with a total of 99 speakers. How many systems are stereo systems and how many are home-theater systems?
17 stereo systems, 13 home-theater systems
Course 3
11-6 Systems of Equations
Learn to solve systems of equations.
Course 3
11-6 Systems of Equations
Vocabulary
system of equationssolution of a system of equations
Insert Lesson Title Here
Course 3
11-6 Systems of Equations
A system of equations is a set of two or more equations that contain two or more variables. A solution of a system of equations is a set of values that are solutions of all of the equations. If the system has two variables, the solutions can be written as ordered pairs.
Course 3
11-6 Systems of Equations
Course 3
11-6 Systems of Equations
When solving systems of equations, remember to find values for all of the variables.
Caution!
Additional Example 1A: Solving Systems of Equations
Solve the system of equations.
y = 4x – 6
y = x + 3
y = 4x – 6 y = x + 3
The expressions x + 3 and 4x – 6 both equal y. So by the Transitive Property they equal each other.
4x – 6 = x + 3
Course 3
11-6 Systems of Equations
Additional Example 1A Continued
To find y, substitute 3 for x in one of the original equations.y = x + 3 = 3 + 3 = 6
The solution is (3, 6).Course 3
11-6 Systems of Equations
Solve the equation to find x.
4x – 6 = x + 3 – x – x Subtract x from both sides.
3x 6 = 3
3x 9 6 6 Add 6 to both sides.
3 = 3 x = 3
Divide both sides by 3.
The system of equations has no solution.
Course 3
11-6 Systems of Equations
2x + 9 = 8 + 2x – 2x – 2x
Transitive Property
Subtract 2x from both sides. 9 ≠ 8
Additional Example 1B: Solving Systems of Equations
y = 2x + 9y = 8 + 2x
Check It Out: Example 1A
Solve the system of equations.
y = x – 5y = 2x – 8
y = x – 5 y = 2x – 8
x – 5 = 2x – 8
Course 3
11-6 Systems of Equations
The expressions x – 5 and 2x – 8 both equal y. So by the Transitive Property they equal each other.
Check It Out: Example 1A Continued
To find y, substitute 3 for x in one of the original equations.
y = x – 5 = 3 – 5 = –2
The solution is (3, –2).
Course 3
11-6 Systems of Equations
Solve the equation to find x.
x – 5 = 2x – 8– x – x Subtract x from both sides.
–5 = x – 8
3 = x
+ 8 + 8 Add 8 to both sides.
The system of equations has no solution.
Course 3
11-6 Systems of Equations
3x + 7 = 6 + 3x – 3x – 3x
Transitive Property
Subtract 3x from both sides. 7 ≠ 6
Check It Out: Example 1B
y = 3x + 7y = 6 + 3x
To solve a general system of two equations with two variables, you can solve both equations for x or both for y.
Course 3
11-6 Systems of Equations
Additional Example 2A: Solving Systems of Equations by Solving for a Variable
Solve the system of equations.
5x + y = 7 x – 3y = 115x + y = 7 x – 3y = 11
y y 3y 3y
Solve both equations for x.
5x = 7 y x = 11 + 3y
15y 15y55 = 7 – 16y
Subtract 15y from both sides.
Course 3
11-6 Systems of Equations
5(11 + 3y)= 7 y
55 + 15y = 7 – y
Additional Example 2A Continued
–7 –7
48 16y
Subtract 7 from both sides.
Divide both sides by –16.
–16 = 16
x = 11 + 3y = 11 + 3(3) Substitute –3 for y. = 11 + –9 = 2The solution is (2, –3).
Course 3
11-6 Systems of Equations
55 = 7 – 16y
3 = y
Course 3
11-6 Systems of Equations
You can solve for either variable. It is usually easiest to solve for a variable that has a coefficient of 1.
Helpful Hint
Additional Example 2B: Solving Systems of Equations by Solving for a Variable
Solve the system of equations.
–2x + 10y = –8 x – 5y = 4–2x + 10y = –8 x – 5y = 4 –10y –10y +5y +5y
Solve both equations for x.
–2x = –8 – 10y x = 4 + 5y
= ––8–2
10y–2
–2x–2
x = 4 + 5y4 + 5y = 4 + 5y
Course 3
11-6 Systems of Equations
5y 5ySubtract 5y from both sides.4 = 4
Since 4 = 4 is always true, the system of equations has an infinite number of solutions.
Check It Out: Example 2A
Solve the system of equations.
x + y = 5 3x + y = –1x + y = 5 3x + y = –1
–x –x – 3x – 3x
Solve both equations for y.
y = 5 – x y = –1 – 3x
5 – x = –1 – 3x+ x + x
5 = –1 – 2x
Add x to both sides.
Course 3
11-6 Systems of Equations
Check It Out: Example 2A Continued
5 = –1 – 2x
+ 1 + 1
6 = –2x
Add 1 to both sides.
Divide both sides by –2.
–3 = x
y = 5 – x = 5 – (–3) Substitute –3 for x. = 5 + 3 = 8The solution is (–3, 8).
Course 3
11-6 Systems of Equations
Check It Out: Example 2B
Solve the system of equations.
x + y = –2 –3x + y = 2x + y = –2 –3x + y = 2
– x – x + 3x + 3x
Solve both equations for y.
y = –2 – x y = 2 + 3x
–2 – x = 2 + 3x
Course 3
11-6 Systems of Equations
+ x + x Add x to both sides.
–2 = 2 + 4x–2 –2
–4 = 4x
–2 – x = 2 + 3x
Subtract 2 from both sides.
Divide both sides by 4.–1 = x
y = 2 + 3x= 2 + 3(–1) = –1 Substitute –1
for x.The solution is (–1, –1).
Course 3
11-6 Systems of Equations
Check It Out: Example 2B Continued
Lesson Quiz
Solve each system of equations.1. y = 5x + 10
y = –7 + 5x
2. y = 2x + 1 y = 4x
3. 6x – y = –15 2x + 3y = 5
4. Two numbers have a sum of 23 and a difference of 7. Find the two numbers.
Insert Lesson Title Here
(–2,3)
15 and 8
( , 2)12
Course 3
11-6 Systems of Equations
no solution