Patterns, Functions, & Algebra – System of Equations (Notes)
Name Date
I can … / Essential Question:Key Concepts / Notes
Two equations together are called:
A solution of a system of equations is an _____________ that satisfies both equations.
A system of two linear equations can have ____, ____, or an ______.
What are the three methods to solve a system of equations?
Types of Solutions:
Number of Solutions
Graph of a System
Meaning
Algebraic Meaning
When is it Best to use the following three ways to solve a system of equations?
1. Substitution
2. Elimination
3. Graphing
Solving a system of equations by substitution:
Steps:
1.
2.
3.
4.
5.
Example 1 – Solve the system of equations using substitution method
x=3-y y=6x-11
x+y=7 -2x-3y=-7
Solving System of Equations by Elimination Method
Goal –
Hint – 1)
2)
But what if they do not have the same coefficient:
When to add the two equations together:
When to subtract the two equations together:
Example 2 – Solving Equations using Elimination Method
4x+y=23
3x-y=12
5x-3y=22
5x-2y=2
3x+4y=6
5x+2y=-4
2x-3y=15
4x-6y=25
When writing equations for a system of equation, it is very important to set up a let statement. A let statement is a statement that describes the variables being used. Try to think about the two things being used, and the two statements that each thing is being used in, these will be the two equations with two unknowns. Let’s try some examples setting up a system of equations and answering them in complete sentences.
Example 3 - The school that Vishay goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. Find the price of a senior citizen ticket and the price of a child ticket.
Example 4 - The state fair is a popular field trip destination. This year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 8 vans and 8 buses with 240 students. High School B rented and filled 4 vans and 1 bus with 54 students. Every van had the same number of students in it as did the buses. Find the number of students in each van and in each bus.
Example 5 - A test has twenty questions worth 100 points. The test consists of True/False questions worth 3 points each and multiple choice questions worth 11 points each. How many multiple choice questions are on the test?
Example 6 – Shaya’na spent $131 on shirts. Fancy shirts cost $28 and plain shirts cost $15. If she bought a total of 7, then how many of each kind did she buy?
Example 7 - Tashana's Custom Kitchen Supplies sells handmade forks and spoons. It costs the store $1.70 to buy the supplies to make a fork and $1.30 to buy the supplies to make a spoon. The store sells forks for $5.60 and spoons for $5.40. Last April Tashana's Custom Kitchen Supplies spent $37.90 on materials for forks and spoons. They sold the finished products for a total of $147.20. How many forks and how many spoons did they make last April? /
Reflection, Summary, & Analysis
Patterns, Functions, & Algebra – System of Equations (Exercise A)
Name Date
Determine the best method to solve the system of equations. Show ALL work.
1. y=-3x+5 2. 3x-4y=-10
5x-4y=-3 5x+8y=-2
3. x=4y+8 4. x-y=2
2x-8y=-3 5x+3y=38
5. 2x-3y=12 6. -3x-3y=3
x+3y=12 y=-5x-17
7. y=3x 8. 6x-y=9
3x+4y=30 6x-y=11
9. 2x-3y=2 10. 5x-2y=12
5x+4y=28 3x-2y=-2
11. 4x+3y=19 12. 2x-y=6
3x-4y=8 3x+4y=-2
Patterns, Functions, & Algebra – System of Equations (Exercise B)
Name Date
1. Juwan and Steve decide to spend the afternoon at an amusement park enjoying their favorite activities, the water slide and the gigantic Ferris wheel. Their tickets are stamped each time they slide or ride. At the end of the afternoon they have the following tickets:Fun Time Amusements
Water Slide:
Ferris Wheel:
Total: $17.70
Juwan's Ticket / Fun Time Amusements
Water Slide:
Ferris Wheel:
Total: $15.55
Steve's Ticket
How much does it cost to ride the Ferris Wheel?
How much does it cost to slide on the Water Slide?
2. There are 13 animals in the barn. Some are chickens and some are pigs. There are 40 legs in all. How many of each animal are there?
3. The senior classes at High School A and High School B planned separate trips to New York City. The senior class at High School A rented and filled 1 van and 6 buses with 372 students. High School B rented and filled 4 vans and 12 buses with 780 students. Each van and each bus carried the same number of students. How many students can a van carry? How many students can a bus carry?
4. Tashana and Shaya’na are selling flower bulbs for a school fundraiser. Customers can buy bags of windflower bulbs and bags of daffodil bulbs. Tashana sold 10 bags of windflower bulbs and 12 bags of daffodil bulbs for a total of $380. Shaya’na sold 6 bags of windflower bulbs and 8 bags of daffodil bulbs for a total of $244. What is the cost each of one bag of windflower bulbs and one bag of daffodil bulbs?
5. A test has twenty four questions worth 100 points. The test consists of extended response questions worth 5 points each and multiple choice questions worth 3 points each. How many of each type of questions are on the test?
6. The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended?
· Two equations together are called ________System of Equations____________.
· A solution of a system of equations is an__ordered pair_____ that satisfies both equations.
· A system of two linear equations can have _0_, _1_, or an ____infinite many______.
· Methods to solve system of equations:
1. Graphing
2. Substitution
3. Elimination
· Types of solutions:
Intersecting Lines / Same Line / Parallel LinesNumber of Solutions / One / Infinite / No Solution/ Zero
Graph of a System / / /
Meaning / Lines only cross in one point. That ordered pair is the solution. / Lines are on top of each other. Intersect at every pt on line. So infinitely many solutions. / Lines never cross, so no ordered pair is a solution
Algebraic Meaning / You actually get an answer. The solution or answer is the ordered pair where the two lines intersect. / The variable will cancel and you will get a true statement. / The variable will cancel and you will get a false statement.
Solving a system of equations by substitution:
Steps:
1. Make sure one of the equations is already solved for a variable.
2. Substitute - Take the equation that is already solved for a variable and plug it into the other equation
3. Solve the equation for the given variable
4. Substitute the value for the variable into either equations and solve for the other variable.
5. Check your answer
Solving System of Equations by Elimination Method
Goal – to eliminate (get ride) either the x’s or the y’s. Then solve the new equation with only one variable.
Hint – 1) Line the two equations over each other with like terms above/below each other
2) Eliminate the variable that has the SAME coefficient.
But what if they do not have the same coefficient:
Multiply either one equation or both equations by a number in order to get the same coefficients.
When to add the two equations together:
When the coefficients are the same but DIFFERENT signs
When to subtract the two equations together:
When the coefficients are the same and SAME signs.