Systems of Equations & Inequalities © Beth MacDonald 2009

Systems – Main MENU What is a system of equations/inequalities? Three types of solutions Methods used to solve Answer the question being asked Dealing with word problems Practice problems Pick from above list to learn more.

What is a system? Two or more equations (or inequalities) create a system. You will be asked to solve at least one system We use systems to solve word problems involving multiple items Return to MENU

Answer the Question Being Asked Carefully read how they want your answer – Ordered pair (x, y) – Sum of the solutions Add x and y together Example: x = 3, y = 1, your answer is then = 4 – Product of the solutions Multiply x and y together EX: x = 3, y = 1, your answer is then 3(1) = 3 – Written in a complete sentence EX: Kathy sold 12 chocolate cakes and 7 vanilla cakes. Return to MENU

Three types of Solutions No SolutionOne Solution (x, y) Infinitely Many Solutions Lines never intersectLines intersect onceLines continuously intersect Same slope, different y- intercept Different slopesSame slope, same y- intercept (same line) y=-x+3 x + y=3 Return to MENU

Three Methods to Solve Solve by GraphingSolve by SubstitutionSolve by Elimination Return to MENU

Solve by Graphing 1.Manually graph both equations on graph paper 2.Use the graphing calculator to graph – Graph a system of equations Graph a system of equations – Graph a system of inequalities Graph a system of inequalities 3. If the point of intersection does not have integers for coordinates, find the exact solution by using substitution or elimination. Return to MENU

Graphing Calculator – pg 1 1.Before you turn on the calculator, solve both equations for y 2. If you’ve never used a graphing calculator, click here to learn about the keys of the calculatorIf you’ve never used a graphing calculator, click here to learn about the keys of the calculator 3. Press ON key (bottom left corner) on your graphing calculator 4. Press Y= key (top left corner) 5. Type 1 st equation into \ Y 1 = 6. Type 2 nd equation into \ Y 2 =

Keys on the graphing calculator Variable x: X,T,θ,n Negative number: (-) key x²: x² key Return to blank screen: 2 nd QUIT erase everything: CLEAR Delete one character at a time: DEL Return to graphing Return to graphing

Graphing Calculator – pg 2 7. Press GRAPH (top right corner) can’t see anything… click HERE to change the viewing windowclick HERE to change the viewing window 8. Press 2 nd CALC (trace key) – Option 5:intersection ENTER – First curve? (move blinker to one of the lines) press ENTER – Second curve? (blinker should have moved to second line) press ENTER – Guess? Press ENTER – Intersection (this gives you the x and y coordinates of the solution)

Graphing Calculator – pg 3 9. How should we answer the question? – Ordered pair solution? – Sum of the solutions? 10. Answer accordingly. Ready to try some… click HEREHERE Return to Solving MENU

Viewing Window Option 1 Press ZOOM Option #0:ZoomFit (last option listed…use arrow key to scroll down) Press ENTER Option 2 Press WINDOW Change the Xmin, Xmax, Ymin, Ymax Press GRAPH

Graphing an Inequality on the calculator 1.Both equations must have y by itself 2.Press ON key (bottom left corner) on your graphing calculator 3.Press Y= key (top left corner) 4.Type 1 st equation into \Y 1 = (the x variable is to the right of the ALPHA key) 5.Move the cursor to the left of \Y 1 = so the \ is blinking and press ENTER until the shading is either up (greater than) or down (less than) 6.Type 2 nd equation into \ Y 2 = 7.Follow step 5 for 2 nd equation 8.Press GRAPH (top right corner) 9.Most likely you’ll be asked to find the ordered pair that is located in the intersection of the two shadings. 10.Answer accordingly. Return to MENU

Substitution Use substitution when one of the coefficients is equal to 1. You’ll substitute part of one equation into the other equation. Once you solve for one variable, you’ll have to use that to solve for the other variable. Return to Solving MENU

Line up your x’s, y’s and equal signs Find a common coefficient for either x or y. Add or subtract your equations to eliminate a variable. Once you solve for a variable, you’ll have to use that to solve for the other variable. Elimination Return to MENU

Word Problems AGHHHHH… the dreaded words… don’t be afraid…and don’t skip them! Read the question carefully Underline what sounds important Try to put yourself into the scenario Create two equations from the given information Pick a method to solve – most likely you’ll use elimination Does your answer make sense? Return to MENU

Practice Problems Return to MENU Practice Problems Return to MENU

Practice Problem #1 What is the sum of the solutions for y = x + 3 3x + y = 5 ANSWER 

Answer 1) What is the sum of the solutions for y = x + 3 3x + y = 5 Return to Practice Problems

Practice Problem #2 Solve. Write your answer as an ordered pair. -0.5x + y = - 1 y - 1 = 2 -7x + 2 ANSWER 

Answer 2) Step 1: solve each equation for y -0.5x + y = - 1 becomes y = 0.5x - 1 y - 1 = 2 -7x + 2 becomes y = -7x +5 Step 2: solve by substitution 0.5x – 1 = -7x x = 6 x = 0.8 Step 3: use y - 1 = 2 -7x + 2, substitution 0.8 for x y – 1 = 2 – 7(0.8) + 2 y = 2 – y = -0.6 Answer: (0.8, -0.6) Return to Practice Problems

Practice Problem #3 Solve the system. What is the sum of x and y? y = 9x + 20 y = -1/3 x + 13 ANSWER 

Answer 3) y = 9x + 20 y = -1/3 x + 13 Use substitution 9x + 20 = -1/3x /3x 9 1/3x + 20 = 13 Return to Practice Problems 9 1/3x + 20 = /3x = /3 x = -3/4 y = 9(-3/4) + 20 y = Sum is Sum is 12.5

Practice Problem #4 Solve. What is the product of x and y? 2y = 3x + 4 y = -2x ANSWER 

Answer 4) Solve. What is the product of x and y? 2y = 3x + 4 y = -2x Answer: -0.5 Return to Practice Problems

Practice Problem #5 Clair bought three bars of soap and five sponges for $2.31. Steve bought five bars of soap and three sponges for $3.05. Find the cost of each item. ANSWER 

Answer 5) Let x = price per bar of soap, y = price per sponge 3x + 5y = x + 3y = 3.05 x = $0.52 y = $0.15 Return to Practice Problems

Practice Problem #6 Kendra owns a restaurant. She charges $1.50 for 2 eggs and one piece of toast, and $.90 for one egg and one piece of toast. Write and graph a system of equations to determine how much she charges for each egg and each piece of toast. Let x represent the number of eggs and y the number of pieces of toast. ANSWER 

Answer 6) Let e = price per egg, t = price per slice of toast 2e + t = 1.50 e + t = 0.90 $0.60 per egg $0.30 for toast Return to Practice Problems

Practice Problem #7 Sharon has some one-dollar bills and some five-dollar bills. She has 14 bills. The value of the bills is $30. Solve a system of equations using elimination to find how many of each kind of bill she has. ANSWER 

Answer 7) Let x = 1 dollar bills, y = 5 dollar bills x + y = 14 x + 5y = 30 4 five-dollar bills 10 one-dollar bills Return to Practice Problems

Practice Problem #8 Tickets to a local movie were sold at $6.00 for adults and $4.50 for students. There were 240 tickets sold for a total of $1, Find the number of adult tickets sold and the number of student tickets sold. ANSWER 

Answer 8) 6a + 4.5s = 1155 a + s = student and 190 adult tickets Return to Practice Problems

Practice Problem #9 Tom has a collection of 27 CDs and Nita has a collection of 18 CDs. Tom is adding 3 CDs a month to his collection while Nita is adding 6 CDs a month to her collection. Write and graph a system to find the number of months after which they will have the same number of CDs. Let x represent the number of months and y the number of CDs. ANSWER 

Answer 9) Tom: y = x Nita: y = x 3 months Return to Practice Problems Months Number of CD’s