Algebraic Word Problems Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher.

Word Problems When given a word problem where you have to find a missing number(s), there are two steps. 1st) Convert the words in the problem to an algebraic equation. 2nd) Solve the equation for x, and use x to find any other missing number(s).

Solving Problems with One Missing Number Ex) Ten less than eight times a number is 22. Find the number. We first need to realize that when the words “a number” are mentioned, this really represents the missing answer that we are finding. You can use the variable x to represent this unknown number. In algebra, what can you call a number that you don’t know the value of?

Solving Problems with One Missing Number Ex) Ten less than eight times a number is 22. Find the number. eight times a number Let’s look at the phrase “eight times a number”. 8x Knowing that we should use the variable x as “the number”, how could we algebraically write eight times a number?

Solving Problems with One Missing Number Ex) Ten less than eight times a number is 22. Find the number. Ten less than Let’s look at the phrase “ten less than”. 8x To make sense of this let’s think about a side example: 10 less than 16 10 less than 16 is 6. What is 10 less than 16 and how would you write that mathematically? Mathematically, that is written as… 16 – 10 = 6

Solving Problems with One Missing Number Ex) Ten less than eight times a number is 22. Find the number. Ten less than With that said, we know that we will have to subtract 10 from 8x just like 10 was subtracted from 16. 8x - 10 8x So now we have the phrase, “ten less than eight times a number” written in algebraic terms.

Solving Problems with One Missing Number Ex) Ten less than eight times a number is 22. Find the number. Ex) Ten less than eight times a number is 22. Find the number. Lastly, let’s look at the phrase “is 22”. 8x - 10 8x - 10 = 22 The word “is” in math means equal. So what would we need to put at the end of 8x-10 to represent “is 22”? = 22

Solving Problems with One Missing Number Ex) Ten less than eight times a number is 22. Find the number. Now that we have our equation written, the last step is to find the number that makes the statement true. In other words, solve the equation for x. 8x - 10 = 22 8x - 10 Let’s use our table in the calculator to do this. So the answer to our problem is… X = 4 Plug 8x – 10 into y = and find 22 in the y column of the table.

Something to Remember 8x - 10 When the problem is… 10 less than 8 times a number 8x - 10 The words “less than” makes the order of the two numbers get reversed.

When the problem is… 10 diminished by 8 times a number Something to Remember When the problem is… 10 diminished by 8 times a number 10 – 8x The words “diminished by” or “decreased by” keeps them in the same order.

When you have the phrase “the sum of a number and …” Something to Remember When you have the phrase “the sum of a number and …” In math, “the sum” means the answer to what mathematical operation? 1st) You can write “the sum of a number and 5” as x + 5 2nd) Since you need to write three times the entire sum, you would multiply that expression by three (with parentheses around “x + 5”) Addition Example: Three times the sum of a number and 5 is 21. What is the number? 3rd) Write what the expression is equal to, to make an equation. 3( ) x + 5 = 21

To solve an equation with x on both sides of the equation… Something to Remember To solve an equation with x on both sides of the equation… Ex) Twice a number is the same as 6 more than the number. What is the number? 1st) Plug the left side of the equation into y1 and plug the right side of the equation into y2 in “y=“ on your calculator. Write an equation for this problem. 2x = x + 6

To solve an equation with x on both sides of the equation… Something to Remember To solve an equation with x on both sides of the equation… Ex) Twice a number is the same as 6 more than the number. What is the number? 2nd) Find where y1 is equal to y2 in the table on your calculator. Write an equation for this problem. 2x = x + 6

To solve an equation with x on both sides of the equation… Something to Remember To solve an equation with x on both sides of the equation… Ex) Twice a number is the same as 6 more than the number. What is the number? 3rd) The x value when this happens is the answer to the problem. Write an equation for this problem. 2x = x + 6 x = 6

Solving Problems with Two Missing Numbers Ex) The first of two numbers is six more than twice the second. The sum of the numbers is 21. Find both numbers. second 1st Number: In this problem we have two numbers, the first number 2nd Number: x and the second number. We need to call one of these two numbers x. To do this, we must look at the last word in the first sentence. So we should use x to represent the second number.

Solving Problems with Two Missing Numbers Ex) The first of two numbers is six more than twice the second. The sum of the numbers is 21. Find both numbers. Now we need to write an expression for the first number. In the problem it says that the first number “is six more than twice the second.” 1st Number: 2x + 6 2nd Number: x Remember that we called the second number x. Take a few seconds and think about how you would write that statement algebraically. 2x + 6

Solving Problems with Two Missing Numbers Ex) The first of two numbers is six more than twice the second. The sum of the numbers is 21. Find both numbers. The second sentence says that the sum of the numbers is 21. What does sum mean in math terms? 1st Number: 2x + 6 2nd Number: x 1st Number 2x + 6 + = 21 2nd Number x Sum means that when you add those two numbers together that they will equal 21. So our equation is… 2x + 6 + x = 21 Let’s write this as an algebraic equation.

Solving Problems with Two Missing Numbers Ex) The first of two numbers is six more than twice the second. The sum of the numbers is 21. Find both numbers. Just like the first example, solve the equation for x using the table in your calculator. 1st Number: 2x + 6 2nd Number: x = 5 2x + 6 + x = 21 So… X = 5 The second number is also represented by x. So the second number is 5.

Solving Problems with Two Missing Numbers Ex) The first of two numbers is six more than twice the second. The sum of the numbers is 21. Find both numbers. We’ve found the second number, but have to find the first number. 1st Number: 2x + 6 = 2(5) + 6 = 16 2nd Number: x = 5 If x is in the expression for the first number, and we know that x = 5, what can we do mathematically to figure out the first number? 2x + 6 + x = 21 So the two numbers that make the statement true are… 16 and 5 We can substitute 5 in for x and compute the expression to figure out the first number.

Solving Problems with Two Missing Numbers Ex) The first of two numbers is six more than twice the second. The sum of the numbers is 21. Find both numbers. Let’s check to see if the sum of these two numbers is 21. 1st Number: 2x + 6 = 2(5) + 6 = 16 2nd Number: x = 5 16 + 5 = 21 2x + 6 + x = 21 21 = 21 ✔ So the two numbers that make the statement true are… 16 and 5 This is a good way to check to see if the two answers that you got are correct.

Solving Problems with Three Missing Numbers Ex) The first of three numbers is triple the third. The second number is 6 more than twice the third. The sum of all three numbers is 54. Find all three numbers. 1st Number: 2nd Number: 3rd Number: Now we have three numbers to find. x Just like the last problem, the number that we will call x should be the last word in the first sentence. What should we call x? The third number should be called x.

Solving Problems with Three Missing Numbers Ex) The first of three numbers is triple the third. The second number is 6 more than twice the third. The sum of all three numbers is 54. Find all three numbers. 1st Number: 2nd Number: 3rd Number: 3x Let’s look at the first sentence now. “The first of three numbers is triple the third.” How can we write this algebraically if the third number is called x? x An expression for the first number is… 3x

Solving Problems with Three Missing Numbers Ex) The first of three numbers is triple the third. The second number is 6 more than twice the third. The sum of all three numbers is 54. Find all three numbers. 1st Number: 2nd Number: 3rd Number: 3x To get the second number we have to look at the second sentence. “The second number is 6 more than twice the third.” What is the expression going to be for the second number? 2x + 6 x An expression for the second number is… 2x + 6

Solving Problems with Three Missing Numbers Ex) The first of three numbers is triple the third. The second number is 6 more than twice the third. The sum of all three numbers is 54. Find all three numbers. 1st Number: 2nd Number: 3rd Number: 3x The next step is to set up the equation using the sentence, “the sum of all three numbers is 54.” How are we going to write this? 2x + 6 x Our equation should be… 3x + 2x + 6 + x = 54 You should get… x = 8 1st Number 2nd Number 3rd Number Now solve this equation for x using a table.

Solving Problems with Three Missing Numbers Ex) The first of three numbers is triple the third. The second number is 6 more than twice the third. The sum of all three numbers is 54. Find all three numbers. 1st Number: 2nd Number: 3rd Number: 3x = 3(8) = 24 Now that we know x = 8, let’s substitute this value into the other expressions for the first, second, and third number. 2x + 6 = 2(8) + 6 = 22 x = 8 So our three missing numbers are… 24, 22, and 8 Let’s make sure their sum is 54. 24 + 22 + 8 = 54 54 = 54 ✔

Follow-Up Questions Answer the following questions on loose leaf and hand them in to your teacher.

Follow-Up Questions Write each equation and solve for the missing number(s). Fifteen less than four times a number is nine. Triple a number decreased by fourteen is sixteen. Twice a number plus 10 is the same as four more than four times the number. The larger of two numbers is three times the smaller number. The sum of the two numbers is 16. The sum of three numbers is 45. The second number is three more than the first number. The third number is twice the second number. Players on a basketball team must buy their entire uniform (jersey, shorts, and shoes). The jersey is twice as expensive as the shorts. The shoes cost eight dollars more than three times the shorts. Each player is charged a total amount of $104. How much did each individual item cost?