SOLVING LINEAR EQUATIONS WITH ONE VARIABLE
The objective for this lesson on Solving Linear Equations with One Variable is, the student will utilize the distributive property and combine like terms in order to solve multi-step equations.
The skills students should have in order to help them in this lesson include: solving two-step equations with one variable.
We will have three essential equations that will be guiding our lesson. Number 1, explain how to identify like terms in an equation. Number 2, why is combining like terms an essential part of solving an equation? Justify your answer. And number 3, how can the distributive property be used to solve equations? Defend your thinking.
We will begin by completing the warm-up solving two-step equations to prepare for solving linear equations with one variable in this lesson.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Matt is buying lunch for his family. Including Matt, there are five people in the family. The total bill for the lunch is thirty five dollars and the cost for each drink is two dollars. Matt keeps twenty dollars in his wallet for emergencies. If each member has a sandwich that costs the same amount, how much is each sandwich?
We will begin by Studying the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, how much is each sandwich?
Now that we have identified the question we want to put this question in our own words in the form of a statement. This problem is asking me to find the cost of one sandwich.
During this lesson we will learn how to solve linear equations with one variable, by combining like terms and applying the distributive properties. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
EXPLORING LIKE TERMS
We are going to begin by exploring like terms. Let’s take a look at the list of terms here. The terms are six a, seven, four b, x, eighteen, four a, three x, and negative two b.
Looking at the list of terms discuss what you notice about how these terms are alike and different. Some of the ways the terms are alike and different: are that some are a variable, some are a variable with coefficient, and some are constants.
How do you think the terms could be combined? Let’s use our algebra tiles to explore how we can work with expressions that have different variables and constants.
Look at the expression in Problem one. The expression is five x plus six minus three x plus seven. What does the variable represent? The variable represents an unknown value that makes the statement true.
Let’s simplify the expression using our algebra tiles. When using the algebra tiles, what do we use to represent the variable or the x-value? We use the long yellow tiles for positive variables or long red tiles for negative variables.
Using the algebra tiles, explain how we can represent the first expression above. The first term in the expression is five x. We will use five long yellow tiles to represent the variable five x.
Our second term is positive six. We will use six yellow unit tiles to represent positive six.
The third term in the expression says that we need to subtract three x. Another way of says we are subtracting three x is to say we are adding negative three x. We can represent adding negative three x with three long red tiles.
And our last term is adding seven. We can represent plus seven with seven yellow unit tiles.
Now that we have represented the expression with the algebra tiles, what do you notice about our expression? We have several long x-tiles and several unit tiles and we have more than one color. How do you think we can simplify the expression? Explain your thinking. Because we are adding, we can push together the tiles that are alike or combine all of the tiles that are the same shape.
What is the name for tiles that have the same shape? These are like terms.
Let’s move all of the tiles so that all of the same-shaped tiles are together. We can put our variable tiles together, and our unit tiles together.
Now are all of the x-tiles or variable tiles the same color? No. Explain what we can do when the tiles are two different colors? We need to cancel out any zero pairs by removing one red and one yellow tile until only one color is left. How many zero pairs should we take away from the variable tiles? We will need to take away three pairs of zero pairs. Let’s do that now. Here’s one pair, two pairs, three pairs. By removing three zero pairs we end up with our variable tiles all being the same color.
How many variable tiles are left? There are two yellow variable tiles, or two x.
Now are all of the unit tiles the same color? Yes, they are all yellow. Let’s count them up. How many unit tiles are there in total? There are thirteen yellow unit tiles.
So now that we have simplified, what is the expression that represents the remaining tiles? The expression is two x plus thirteen. We have two yellow variable tiles and thirteen yellow unit tiles.
Can we say that this expression is now simplified? Justify your answer. The expression is simplified because there are no more like terms that can be combined.
Now on your page the left side of the box will be used to represent the expression pictorially for each problem.
Let’s represent what we did with our algebra tiles pictorially in this box. What was the first step that we did to simplify the expression when we were using the algebra tiles? We represented the expression with five yellow variable tiles, six yellow unit tiles, three red variable tiles and seven more yellow unit tiles.
Let’s draw these tiles in the box on the left. We need five yellow variable tiles, six yellow unit tiles, three red variable tiles and seven more yellow unit tiles.
Now what was the next step that we did with the algebra tiles? We combined all of the tiles that are the same size and identified whether or not they were all one color. Let’s draw the sorting of the tiles below the first picture. We will draw the variable tiles together and then draw the unit tiles together.
When we were using our algebra tiles, what was the last step we did? We canceled the zero pairs of the variable tiles by removing one red and one yellow variable tile until just one color remained. To show this in our pictorial representation, we need to cross out the zero pairs on the pictorial representation and draw the final representation of the tiles that are left over. Let’s do that now. We removed three zero pairs from our variable tiles to represent this pictorially we will draw an x over these three sets of zero pairs.
What remains is, two yellow variable tiles and thirteen yellow unit tiles. What is our simplified expression? Since the expression contains two x-tiles and thirteen unit tiles, we can write this expression as two x plus thirteen.
Let’s record this answer in our chart. Five x minus three x equals two x.
Now let’s look at the constants. Can we combine the constants? Yes we can. What is positive six plus positive seven? It is positive thirteen. Let’s record this in our chart.
Now what do you notice about the simplified columns? The total at the bottom of the column represents the simplified expression we found earlier using the algebra tiles. The expression is two x plus thirteen.
Now take a look at the bottom right box on your page. The bottom right box will be used for an abstract method of simplifying. Copy the original expression into this box.
What was the next step pictorially that we did to simplify? We combined all of the like terms by grouping them together.
Let’s rewrite the expression so that the terms with the variable are first and the constants are second. When we rewrite the expression we will have five x minus three x plus six plus seven. We put the like terms together.
What do you notice about the operation between five x and three x? It is subtraction.
What was the next step when we represented the expression pictorially? We cancelled the zero pairs, which subtracted three x from five x. What was left over after we took away the zero pairs? We were left with two x. So what is left if we subtract five x minus three x? Two x.
What process did we use for combining the like terms? We add or subtract the coefficients, depending on if the operation is addition or subtraction between the terms.
What was the last step we did with our algebra tiles, and in our pictorial representation? We added the constants together: six plus seven equals thirteen.
So what is the final simplified expression? Two x plus thirteen. Let’s record this in the box for our abstract representation. The simplified expression is two x plus thirteen.
Now let’s discuss our conclusions. The expressions we worked with had several terms. How were we able to simplify them? We combined the like terms.
Let’s go back and look at our list of original terms from the beginning of this portion of the lesson. Can any of the terms be combined? Yes, we can combine like terms.
Let’s do so now. We can combine six and four a. Since both are positive we will add these together. Six a plus four a equals ten a. We can also combine the constants, seven and eighteen. Again, since both are positive we will add these together. Seven plus eighteen equals twenty five. We can also combine our terms that contain the variable b. Since one of these terms is negative, we will use subtraction. Four b minus two b equals two b. And finally we can combine terms with the variable x. X and three x are these terms. Since both are positive we will add these terms together. X plus three x equals four x.
Now think about this. If we added the term three x squared to the list, could we combine it with the other x-terms? No we couldn’t. Explain your thinking. When we combine like terms, the variable must have the same exponent. With this new term three x squared the exponent is two. We do not have any other terms in our list that have an exponent of two on our x-variable. So this term could not be combined with any of the other terms.
SOLVING EQUATIONS BY COMBINING LIKE TERMS
We are now going to apply what we have learned about combining like terms to solve equations. When we are solving equations, what are the two goals? We need to isolate the variable and balance the equation.
Take a look at the first equation. It is four x plus five minus three x – minus three equals four. What makes up this equation? It is two expressions that are set equal to each other to create an equation.
What did you learn about the terms in an expression in the previously completed activity? Defend your thinking? We learned that we can look for like terms to simplify the expression.
What is the first expression in the equation? It is four x plus five minus three x minus three. How can we represent this expression using tiles? We need to represent each term in the expression. When we have a minus sign before a term in the expression we can represent this by using the opposite sign of the term. So for instance minus three x can be represented by using negative three x in our algebra tiles. So for this expression we would use four yellow x-tiles to represent four x, five yellow unit tiles to represent positive five, three red x-tiles to represent subtracting three x or negative three x, and three red unit tiles to represent subtracting three or negative three.
Let’s use the tiles to represent this on the left side of the balance scale. We have four yellow x tiles, five yellow unit tiles, three red x tiles, and three red unit tiles.
What is the expression on the other side of the equal sign? It is four. How can we represent the four using the unit tiles? We can use four yellow unit tiles to represent four. Let’s do so now on the balance scale on the right side of the balance.
Now let’s sort the like tiles placing the x-tiles together and the unit tiles together. Remember that one of goals when solving equations is to balance the equation. So we do not want to move our tiles from one side of the equation to the other side.
The unit tiles on the left side of the equation can be grouped together. But the unit tiles on the right side of the equation need to stay on the right side of the equation.
Now looking at the expression on the left, is it possible to simplify the expression? Justify your answer. Yes, we can combine like terms by canceling zero pairs. Do we have zero pairs for the x-tiles? Yes, there are three zero pairs. Let’s remove the zero pairs now. Take a look at the unit tiles. Do we have zero pairs for the unit tiles on the left side of the equation? Yes, there are three zero pairs for the unit tiles. Let’s remove those zero pairs now.
Can we simplify the expression on the right side of the equation? No. So what is the simplified equation? It is x plus two equals four. Because we have one long yellow tile on the left hand side along with two yellow unit tiles. And on the right hand side of the equation we have four yellow unit tiles. So this gives us the simplified equation of x plus two equals four.
Can the expression on either side of the equal sign be simplified any further? No, because there are no like terms. So how can we describe the equation we have now? It is a one-step equation.
What is the goal at this point? Our goal after combining like terms is to solve for the variable, x, by isolating it. How can we isolate the variable? We need to get the variable by itself. To do this we can take away two yellow unit tiles from each side. Let’s do that now. What is remaining on our balance scale after we remove the two yellow unit tiles from each side? We are left with x equal to two.
Let’s create a pictorial representation of this equation. What was the first step? We represented the equation with four yellow variable tiles, five yellow unit tiles, three red variable tiles and three red unit tiles on one side of the equal sign and four yellow unit tiles on the right side of the equal sign.
Let’s draw this representation. Now what was the next step? We combined all of the tiles that are the same size and identified whether or not they were all in one color. Let’s draw the pictorial representation for combining the like terms on each side of the equation. We combine the x terms on the left hand side and the constants on the left hand side of the equation. On the right hand side of the equation there is nothing to combine. So we will copy the same four unit tiles on the right hand side of the equation that we had before.
Now what was the next step? We canceled out the zero pairs of the variable tiles. We had three sets of zero pairs for the variable tiles that we can cancel out. We also had three set of zero pairs for the unit tiles that we can cancel out. After doing this we end up with x plus two equals four.
Are there any like terms to be combined or zero pairs on the right side of the equation? No.
So explain what your picture looks like after we simplify. We have one yellow variable tile and two yellow unit tiles on the left side of the equal sign; and we have four yellow unit tiles on the right side of the equal sign.
What tile represents our variable? The long yellow tile represents our variable. So explain how we can isolate that tile. When we isolated that tile in our concrete representation, we removed the two unit tiles from the left side of the equation and from the right side of the equation. To show this pictorially we will cross out the two unit tiles to represent the subtraction. Remember that we need to keep our scale balanced, so we will subtract two unit tiles from each side of the equation. We show this by crossing them out in our picture.