chapter 7 notes

Algebra I Chapter 7 Mrs. Cataldo Class Notes Chapter 7.1 -7.7 Short Quiz: Fill in the chart. Use the equation 6y + 2x = 8. Show work. Starting equation Standard form a,b,c Slope-intercept form Slope, y-intercept Graph the equation. What kind of function is it?__________________ 7.1 Simultaneous Equations When you work with ______ equations at the same time to solve for x and y they are called _____________________________ equations. The general rule of thumb for solving algebraic equations is that for every variable you have to solve for, you need one equation. So, if you want to solve for x and y, two variables, you need ____ equations. If you want to solve for x, y, and z, three variables, you need _____ equations. In the past we have been solving for x with one equation like when we have the sides of a polygon given in terms of x, and the perimeter in terms of x. Then we write one equation and solve for x. When we have a linear equation like y = 3x + 4, we can choose any x and solve for y that will make the sentence true. We have in essence, two equations.: y= 3x + 4 and x = 6. So we can solve for y, y = 22. Now we will work with two equations to find an x,y pair that is a solution to _______. In your book, we have a very tall man and a very short man. We want to find their heights if the sum of their heights is 131 inches and difference between their heights is 65 inches. Strategy: 1. What do we want to know? Assign each unknown a variable. The _______ of the tall man = _____ The _______ of the short man = ______ 2. What do we know? Can we write two equations? a. The sum of their heights is 131 inches. _____________________________ b The difference between their heights is 65 inches. _____________________________ Now we need to solve for x and y. What values of x and y will satisfy both equations? There are various ways to solve these equations. We will learn a few. First, you can actually ______ both equations. Remember the balance. Both sides of an equals sign are equal, they balance. They represent the same values on both sides. So adding one equation to another is the same as adding the same value to both sides of the equation. x + y = 131 + x – y = 65 2x + 0 = 196 Do you see that we have eliminated the y variable so that we can now solve for x? x = Now we can go back to one of the original equations and plug in x and solve for y. Does it matter which equation we pick? ______ ___ + y = 131 y= _____ We can plug x in the other equation and also solve for y. ____ -y = 65 y = _____ Is this the same answer? We write the solution as an ordered pair, ( ____, _____) CHECK your answer!!! NOTE: When you add equations, you must be careful to only add like terms. To make sure you do this you can write the equations so that the like terms are directly below each other. Example: p. 288 #6 First we have to write ____ equations using the graphs given. ____x + ____ y = _____ 14 + y = ____ x Next we need to rearrange the second equation so that the like terms are beneath the first equation. Add equations: 14 +y = 3x 3x + y = 22 -14 -14 + 3x + -y = 14 y = 3x -14 6x + 0 = 36 -3x -3x (-1) (-3x +y)= (-14)(-1) x = 3x +-y = 14 Now substitute x in the first equation and solve for y. 3(___) + y = 22 y = ____ Write the solution as an ordered pair: __________ 7.2 Subtracting Equations If you can add equations, you can also ________________ them. This is tricky. You must keep track of the signs in the second equation. In order to do this, I prefer to add the opposites like we were doing before. In the book you have the following problem with three cats and one kitten weighing 24 pounds and three cats and five kittens weighing 36 pounds. How can you find the weight of the cats and kittens? STRATEGY: 1. What do you want to know? Assign them variables. 2. What do you know? Can you write two equations to solve for both variables? 3. Can you eliminate one variable by adding the equations? _____ 4. Can you eliminate one variable by subtracting the two equations? ______ 3x + y = 24 -(3x + 5y = 36) y = ________ 0x -4y = -12 Subtract each term OR write as an addition problem. 3x + y = 24 y = ________ + -3x -5y = -36 0x + -4y = -12 5. Solve for x by substituting the y in the first equation. Write your solution as an ordered pair. 6. CHECK your answers!!!! 7.3 More on Solving by Subtraction and Addition Review: In order to solve for two variables, you need ____ equations. One strategy you can use to solve for one variable is to add the equations. Another similar strategy is to __________ one equation from the other. This is also called adding the _________ of the second equation. 4x + 7y = -10 4x + 7y = -10 -(4x + y = -22) + -4x + -y = 22 Also, the goal of adding or subtracting two simultaneous equations is to get rid of one variable. What do you do when addition or subtraction alone will not eliminate one variable? 4x + y = 28 ? 2x + 3y = 24 Answer: Multiply or divide one of the equations so that the number of x’s or the number of y’s is the same and can be eliminated with addition or subtraction. In the above case, multiply the second equation by 2 or the first equation by 3. Let’s try both and see what happens. Examples from p. 302 11. 6x + 6y = 24 10x -y = -15 12. 5x – 7y = 54 2x – 3y = 22 13. 1.5x + 2.5y = 16 3x -1.5y = -33 7.4 Graphing Simultaneous Equations Another way to solve simultaneous equations (to come up with an x,y pair that will make both equations true), is to graph both linear functions and see where they intersect. Let’s find the solution to the following simultaneous equations. x + y = 6 y = 2x Plot them both. Find where they intersect. Answer: ( , ) Examples from p. 309 11. 3x – 2y = 6 y = -x – 3 12. y = 7 – x x – y = 2 Short Quiz: 1. Solve ax + by = c for y. 2. Find the length of this rectangle’s sides if the perimeter is 36 and the area is x + 36. 11 3. Solve these simultaneous equations. x + 2y = 10 2x – 3y = 6 7.5 Inconsistent and Equivalent Equations So far we have looked at linear equations that intersect in one place and therefore have one solution, (x,y), that satisfy both equations. Sometimes when we are asked to solve simultaneous equations that look quite different, we can graph them and they never intersect. Two lines that don’t intersect are _________________________. Graph x + y = 5 and y = -x + 3. What is the solution to these equations? ________ When two linear functions don’t intersect they are _____________________. By rewriting the equations, we can see this is true. y = -x + 3 can be written as x + y = 3 by adding x to both sides. Can x + y = 5 and x + y = 3 be true at the same time? ____ Subtract x + y = 5 x + y = 3 This solution is never true. A third thing can happen when solving two equations. Graph x + y = 5 and y = -x + 5. What happened? ______________ When two equations are the same function they are _______________ How many solutions do these equations have? __________ Subtract x + y = 5 x + y = 5 These are always true. Look at page 314 # 4 and describe a, b and c. Do page 315 #8. 7.6 Solving by Substitution We have developed two basic strategies for solving simultaneous equations, ADDING (or subtracting) the equations, and GRAPHING them to find their intersection. A third, most obvious way to solve simultaneous equations is to solve one for either x or y and plug that solution into the first equation to solve for the opposite variable. Then you can solve for the original one. This is called _______________________________. Page 319 in your book shows Snoopy and the bird on the see saw. In science class we learned that the force on one side of the see saw equals the mass of the person or thing on the see saw times the distance from the weight to the fulcrum (the balance point). In order for two people of different weights to balance, they must adjust their distance to the center. If Snoopy weighs 22 lbs and the bird weighs 2 pounds and they sit 12 ft apart, how far from the fulcrum do they each have to sit to balance? What do we want to know? Let x be the distance from the fulcrum for Snoopy Let y be the distance from the fulcrum for bird What do we know? 22x = 2y weight1 • distance1 = weight2 • distance2 the forces must be the same to balance x + y = 12 The distance between them is 12 ft . We could solve by other methods but let’s solve the first for y. 22x = 2y 2 2 11x = y Substitute 11x into the first equation for y. x + 11x = 12 and solve for x. 12x = 12 x = 12/12 x =1 Plug this in to one of the equations and solve for y. 11(1) = 11 y = 11 So Snoopy has to sit 1ft from the fulcrum and bird 11 feet from the fulcrum. The solution can be written as ( , ) Example: Do problem #10 page 324. y = 7x + 10 y = 4 + x 7.7 Mixture Problems First do the Set 1 problems on page 330 for review. Attach the problems to your notes. In this section we will use two equations to solve story problems where we have two unknown variables. Archimedes wanted to find out if a crown the king had was really pure gold as its maker claimed. 1. What did Archimedes want to know? He wanted to know how much of the crown was gold and how much was silver (he knew it could be part silver). Let x be the weight of gold in the crown Let y be the weight of silver in the crown 2. What did Archimedes already know? He knew that 1kg of gold had a volume of 50 cubic centimeters (cm³) and 1 kg of silver had a volume of 100 cm³. So x kg of gold has a volume of 50x cm³ and y kg of silver had a volume of 100y cm³. The actual volume of the crown in question is 140 cm³. 50x + 100y = 140 He also knows the weight of the crown in question is 2 kg so, x + y = 2 3. Solve both equations together to get y and then x as before. 50(x + y) = 2(50) y = 40/50 = 0.8kg 50x + 50y = 100 -(50x + 100y =140) x + 0.8 = 2 so x = 1.2kg -50y = -40 -50 -50 The crown is not pure gold. We’ll do problem 5 on page 330 together.