Sec 3.1 Systems of Linear Equations in Two Variables
Learning Objectives:
1. Deciding whether an ordered pair is a solution.
2. Solve a system of linear equations using the graphing, substitution, and elimination method.
1. Deciding Whether an Ordered Pair Is a Solution
System of Equation—consists of at least two or more linear equations.
Example.
=
−
−
=
−
0 2
12 3 . 4
1 x y
y
x
= +
=
− +
= +
− 0 2
2 2
5 .
2
z x
z x y
z y x
Solution of the system—is the point(s) where the graphs intersect (give true for both equations) Example 1. Determine whether the ordered pairs are a solution to the system:
= +
=
−
6 2 3
4 2
y x
y x
a. (2,0)
Answer:___________________
--- b. (4,−3)
Answer:___________________
--- 2. Solve a system of linear equations using the graphing method
Three types of the System of Equations.
1. Consistent system with Independent equation
• Two lines intersect at one point( )x,y .
• Has one solution,( )x,y .
• m ≠1 m2
• When solve the system, get x = a number, y = a number.
2. Inconsistent System
• Two lines are parallel.
• Has no solution, φ or { }.
• m =1 m2 and b ≠1 b2
• When solve, get false statement.
3. Consistent system with dependent equation
• Two lines lie on top of the others (same line).
• Has infinitely many solutions,
{
(x, y)y=mx+b}
or{
(x, y)ax+by=c}
• m =1 m2 and b =1 b2
• When solve the system, get true statement.
x y
x y
x y
Steps to solve linear equations by graphing 1. Solve and graph each equation separately.
2. Identify type of systems (consistent, inconsistent, or dependent).
3. State number of solution (one solution, infinitely many solutions or no solution).
Example 2. Solve by graphing. Label at least two points for each graph on the graph grid.
=
−
= +
0 4 2
4 2
y x
y x
Solution:___________________
--- 3. Solve a system of linear equations using the elimination method
Steps:
1. Write each equation in the form: Ax+By=C 2. Choose variable to eliminate.
3. If necessary, multiply one or both equations by appropriate number(s) so that the coefficients of the eliminated variable will have the sum of zero.
4. Add two equations together.
5. Solve for the variable.
6. Solve for the other variable.
7. State the final solution in ordered pair, if it exists.
Example 3. Solve linear equations using the elimination method.
−
=
=
−
− y x
y x
6 4 8
0 4 3 2
Answer:___________________________________
--- 4. Solve a system of linear equations using the substitution method
Steps:
1. Solve one of the equations for one of its variable: x or y.
2. Substitute the resulting found in step 1 into the other equation.
3. Solve the equation found in step 2 to find the value of one variable.
4. Substitute the value found step 3 in any original equations containing both variables to find the value of the other variable.
5. Check the solution by substituting the numerical values of the variables in both original equations.
–5 5
5
–5
Example 4. Solve linear equations using the substitution method.
+
=
=
−
7 14 4
7 7 2
y x
y x
Answer:___________________________________
--- Example 5. Solve linear equations using either substitution or elimination method.
1.
−
=
−
−
=
6 3 4
3 2 4
y x
x y
Answer:___________________________________
--- 2.
−
= +
−
=
−
4 2 5
5 2 5 1 2 1
y x
y x
Answer:___________________________________
Sec 3.2 Problem Solving: Systems of Two Linear Equations
Learning Objectives:
1. Use a system of equations to solve problems.
Problem-Solving Steps:
1. UNDERSTAND the problem by do the following:
• Read and reread the problem.
• Identify what is given and what is the question.
• Choose two variables to represent the two unknowns being asked.
• Construct a drawing if needed.
2. TRANSLATE the problem into two equations.
3. SOLVE the system of equations.
4. INTERPRET the results: Check the proposed solution in the stated problem and state your conclusion.
1. Finding Unknown Numbers
Solve the following problems by (a) Choose the variables to represent the unknown. (b) Set up a system and solve. (c) Write the answer using a complete sentence.
Example 1. The sum of two numbers is 56. Their difference is 12. What are the numbers?
First Number is____________________
Second Number is___________________
--- 2. Uniform Motion Problems:
Formula: d = rt where d = distance; r = rate or speed; t = time
Example 2. With a tailwind, a small Piper aircraft can fly 600 miles in 3 hours. Against this same wind, the Piper can fly the same distance in 4 hours. Find the speed of the Piper and speed of the wind.
Piper speed is____________________________
Wind speed is___________________________
t × r = d
With Wind × =
Against Wind × =
3. Solving a Problem about Prices:
Formula: Total price = Number of tickets × price per ticket
Example 3. Admission prices at a local weekend fair were $ 5 for children and $7 for adults. The total money collected was $3379, and 587 people attended the fair. How many children and how many adults attended the fair?
Numbers of tickets × Price per ticket = Total price
children × =
adults × =
There were________________adults and ______________children attended the fair.
--- 4. Coin Problems
Formula: Total Value = numbers of coins× value of each coin
Example 4. Tim has $ 1.10 in quarters and nickels. How many quarters and nickels does he have if he has 14 coins in total?
Tim has _______________quarters and ________________ nickels.
Numbers of coins × Value of each coin = Total value
quarters × =
nickels × =
5. Investment Problems
Formula: I=Prt
Where I = interest earn, P = principal, r = interest rate, t = time (in year)
Example 5. Lit invested $6000, part at 6% and the rest at 4%. How much is invested at each rate if the annual income from the two investments is $290?
P × r × t = I= interest
Account 1 × × =
Account 2 × × =
Lit invested________________________at 6% and ______________________at 4%
--- 6. Mixture Problems:
Formula: Amount of solution = number of liters × percent of the solution Example 6. Nancy wants to make 50 liters of a 60% alcohol solution. She currently has a 20% alcohol solution and a 70% alcohol solution. How many liters of a 20% alcohol solution and a 70% alcohol solution she needs to make 50 liters of a 60% alcohol solution?
Number of liters × Percent of solution = Amount of solution
Solution 1 × =
Solution 2 × =
Mixture × =
Nancy needs _____________________liters of 20% and ____________________liters of 70%
Sec 3.3 Systems of Linear Equations in Three Variables
Learning Objectives:
1. Solve systems of three linear equations containing three variables.
2. Model and solve problems involving three linear equations containing three variables.
1. Solve systems of three linear equations containing three variables Definitions:
1. Linear Equations in Three Variables—Algebraic equation of the formax+by+cz=d, where a, b, c and d are real numbers, with a, b and c are not all zero.
2. A Solution to a system of equation is any ordered triple (x, y, z)that give true statement to all equations in the system.
Example 1. Determine whether the given ordered triple (−1,2,−3) is solution of the system of linear equation.
−
= +
−
=
− +
−
= + +
9 2
2
12 3 2
2
z y x
z y x
z y x
Answer:___________________________
--- Three types of the system
1. Consistent System with independent equations (independent system)-has exactly one solution(x, y, z).
2. Inconsistent System-has no solution, φ.
3. Consistent System with dependent equations (dependent system)—has infinitely many solutions.
Steps for Solving Systems of Linear Equations in Three Variables
1. Select two of the equations and eliminate one of the variables form one of the equations. Select any two other equations and eliminate the same variable from one of the equations.
2. You will have two equations that have only two unknowns. Eliminate a second variable form the two linear equations in two unknown.
3. Solve the remaining variable.
Example 2. Solve each system of equations.
1.
= +
= +
=
− 8
7 2 2
3
z y
z x
y x
Answer:_____________________________
---
2.
=
− +
−
=
− +
= +
−
2 3
5 2
14 3 2
z y x
z y x
z y x
Answer:_____________________________
---
3.
= + +
−
= +
−
−
=
− +
4 2
7 3
2
3 2
z y x
z y x
z y x
Answer:_____________________________
2. Model and solve problems involving three linear equations containing three variables Example 3. Curve Fitting
The function f( )x =ax2 +bx+c is a quadratic function, where a, b, and c are constant.
a. If f( )1 =4, then 4=a( )12 +b( )1 +c or a+b+c=4. Find two additional linear equations if
( )−1 =−6
f and f( )2 =3.
System:__________________________________
--- b. Use the three linear equations found in part (a) to determine a, b and c. What is the
quadratic function that contains the points (−1,−6) ( ), 1,4 and (2, 3)?
a = _______________
b = _______________
c = _______________
Function:__________________________________
Sec 3.6 Systems of Linear Inequalities
Learning Objectives:
1. Graph a linear inequality in two variables.
2. Determine whether an ordered pair is a solution to a system of linear inequalities.
3. Graph a system of linear inequalities.
1. Graph a linear inequality in two variables Definition:
Linear Inequality—is an equation of the form ax+by<c; ax+by>c; ax+by≥c; ax+by≤c Steps to solve linear inequality:
1. Solve inequality in the form y>mx+b; y<mx+b; y≥mx+b; y≤mx+b 2. If inequality involving ≤ or ≥, draw a solid line.
If inequality involving < or >, draw a dashed line.
3. Pick a test point. Substitute the values in the inequality.
If the result is true, shade the side that contain the test point.
If a false statement, shade the other side.
Shaded below the line if y<mx+b or y≤mx+b
Shaded above the line if y>mx+b or y≥mx+b
CAUTION! If multiply or divide by a negative number, the inequality sign change to opposite Example 1. Graph the following inequality. Label at least two points on the graph grid.
6 3
2x− y<−
--- 2. Determine whether an ordered pair is a solution to a system of linear inequalities
Solution to a system of linear inequalities is an ordered pair that satisfies a system of linear inequalities. (It makes each inequality in the system a true statement)
Example 2. Is (−3,−6) a solution to the system of linear inequalities?
≤
−
≤ +
9 3 2
8 2
y x
y
x ;
Answer:_____________________________
–5 5
5
–5
3. Graph a System of linear inequalities Recall:
1. Using solid line when the inequality is nonstrict ( ≤ or ≥ ) 2. Using a dashed line when the inequality is strict (< or > )
3. If multiply or divide by a negative number, the inequality switches.
4. The solution of the system of the inequalities is the overlap portion. If there is no overlap, then the system of inequalities has no solution.
Example 3. Graph the system. State the corner points and tell whether the graph is bounded or unbounded.
1.
−
≤ +
>
+ 2 2
3 2
y x
y x
Answer:____________________________ Graph:_________________________ Corner points:_____________________________
---
2.
≥
≥
≤ +
<
+
0 0
4 2
3
y x
y x
y x
Graph:_________________________
Corner points:________________________________________
–5 5
5
–5
–5 5
5
–5
Example 4. Write a system of linear inequalities that has the given graph.
System:________________________________________
Notes: