Multi-Step Equations How to Identify Multistep Equations |Combining Terms| How to Solve Multistep Equations | Consecutive Integers | Multistep Inequalities

How to Identify Multistep Equations Some equations can be solved in one or two steps Ex) 4x + 2 = 10 is a two-step equation Subtract 2 Divide both sides by 4 Ex) 2x – 5 = 15 is a two-step equation Some equations can be solved in one step, and some require two. The equation 4x + 2 = 10 {four x plus two equals ten} is a two-step equation. Solving it requires first subtracting 2 from both sides and then dividing both sides by 4. Although this equation requires two steps to solve, there is only one unknown term (x). The equation 2x – 5 = 15 {two x minus five equals fifteen} is another example of a two-step equation. The variable x appears only once, and solving for this variable requires only two steps.

How to Identify Multistep Equations Multistep equation – an equation whose solution requires more than two steps Ex) 5x – 4 = 3x + 2 and 4(x – 2) = 12 Multistep equations can take different forms Variables present in two different terms Ex) 6x – 2x = 8 + 4, 5x – 4 = 3x + 2 A multistep equation is an equation whose solution requires more than two steps. One example is the equation 5x – 4 = 3x + 2 {five x minus four equals three x plus two}. Notice that although there is only one unknown (x), it is present on both sides of the equation. These two occurrences of the variable can be combined, but doing this requires extra steps. The equation 4(x – 2) = 12 {four times x minus two equals twelve} is another example of a multistep equation. In this example, the distributive property must be used in order to simplify the left side of the equation. As a result, this equation requires more than two steps to solve. Although multistep equations can take different forms, there are certain features that indicate when an equation requires multiple steps in order to solve. One is if the variable is present in two different terms. These terms could be on the same side of the equation, as in 6x – 2x = 8 + 4 {six x minus two x equals eight plus four}, but frequently they are on opposite sides, as in 5x – 4 = 3x + 2 {five x minus four equals two x plus two}. Another feature that can indicate a multistep equation is the presence of parentheses on either side of it, as in 4(x – 2) = 12 {four times x minus two equals twelve. Some characteristics that indicate that an equation may require more than two steps are listed. Characteristic Example Multiple variable or constant terms on the same side x + 2x + 3x – 1 = 3 + 20 Variable present on both sides 4x – 2 = 3x + 3 Parentheses present on either side 3(x + 2) = 21

Combining Terms First step to solving a multistep equation is to simplify each side Use the distributive property to eliminate parentheses Ex) 5(x – 2) + x = 2(x + 3) + 4 simplifies to 5x – 10 + x = 2x + 6 + 4 Combine like terms Ex) 5x – 10 + x = 2x + 6 + 4 Combine 5x and x to 6x on the left side Combine 6 and 4 to 10 on the right side Simplifies to 6x – 10 = 2x + 10 Terms containing variables cannot be combined with constant terms The first step in solving a multistep equation is to simplify each side. This means using the distributive property, if necessary, to eliminate parentheses. Both sides of the equation 5(x – 2) + x = 2(x + 3) + 4 {five times x minus two plus x equals two times x plus three plus four} contain parentheses that can be eliminated in this way. The distributive property simplifies this equation to 5x – 10 + x = 2x + 6 + 4 {five x minus ten plus x equals two x plus six plus four}.   Simplifying each side also involves combining like terms. On the left, the terms 5x and x can be combined to 6x since each is a multiple of the same variable. On the right, the constant terms 6 and 4 can be combined. This produces the simplified equation 6x – 10 = 2x + 10 {six x minus ten equals two x plus ten}. It is important to remember that terms containing the variable cannot be combined with constant terms.

Combining Terms Consider 5(x – 3) + 4 = 3(x + 1) – 2 Distributive Property can be used on both sides due to the parentheses multiplied by constants Produces 5x – 15 + 4 = 3x + 3 – 2 Terms can be combined on both sides Combine –15 and 4 on the left and 3 and –2 on the right Produces 5x – 11 = 3x + 1 Another example of a problem that requires combining like terms is the equation 5(x – 3) + 4 = 3(x + 1) – 2 {five times x minus three plus four equals three x plus one minus two}. The numbers multiplied by expressions in parentheses make it clear that the distributive property must be used on both sides of this equation. Using the distributive property on each side produces 5x – 15 + 4 = 3x + 3 – 2 {five x minus fifteen plus four equals three x plus three minus two}. This new equation has terms on each side that can be combined, the –15 and 4 on the left and the 3 and –2 on the right. This simplifies the equation to 5x – 11 = 3x + 1 {five x minus eleven equals three x plus one}.

How to Solve Multistep Equations Consider the equation 4(x + 2) – 10 = 2(x + 4) Simplify with the distributive property 4x + 8 – 10 = 2x + 8 Combine like terms 4x – 2 = 2x + 8 Eliminate the unknown from one side 2x – 2 = 8 Eliminate the constant term on the other side 2x = 10 Divide each side by the coefficient of the variable x = 5 Multistep equations can usually be solved by a straightforward process. The first step is to simplify each side by eliminating grouping symbols and combining like terms. Consider the equation 4(x + 2) – 10 = 2(x + 4) {four times x plus two minus ten equals two times x plus four}. The presence of parentheses, as well as the apparent complexity of the equation, indicates that this is a multistep equation. Each side can be simplified by the distributive property, producing the form 4x + 8 – 10 = 2x + 8 {four x plus eight minus ten equals two x plus eight}. The left side can now be simplified further because it contains two constant terms. Combining these simplifies the equation to 4x – 2 = 2x + 8 {four x minus two equals two x plus eight}.The equation is now simpler in form than the original, but the unknown (x) still appears on both sides. It is easier to solve for the unknown if it occurs only once in the equation. As a result, the second step of solving a multistep equation is to eliminate the unknown from one side of the equation. It can be eliminated from either side, but since the variable is traditionally written on the left, it is more common to eliminate it from the right side of the equation. In this example, subtracting 2x from each side of the equation eliminates the variable from the right side, leaving 2x – 2 = 8. This is now a two-step equation since the variable occurs only once, and only two operations are performed on it, namely multiplication and subtraction. As a result, it can now be solved as a two-step equation by undoing each of these operations. The next step is to eliminate the constant term on the side with the variable. Adding 2 to both sides yields 2x = 10. The final step is to divide both sides by the coefficient of the variable. Dividing each side of this equation by 2 results in the solution, x = 5.

Steps to Solve a Multistep Equation How to Solve Multistep Equations Same general set of steps is useful in solving many multistep equations Steps to Solve a Multistep Equation First step Eliminate all variables from the right side (move left) Second step Eliminate all constants from the left side (move right) Third step Simplify each side as much as possible Fourth step Divide each side by the coefficient of the variable Solving this equation required only four basic steps. Although multistep equations can appear in many different forms, the same general set of steps is useful in solving many of them. After identifying the problem as a multistep equation, apply the following steps. First, simplify each side of the equation as much as possible by combining like terms and using the distributive property if necessary in order to remove parentheses. Second, eliminate the variable on one side of the equation. Third, eliminate the constant term on the side with the variable. Fourth, divide each side by the coefficient of the variable. Some multistep equations may not require every step. For example, if the variable has no coefficient after the second step is completed, then the fourth step is unnecessary.

How to Solve Multistep Equations Example Ex) Mark and Susan are given the same amount of money. Mark spends $5, and Susan spends $20. If Mark now has twice as much money as Susan, how many dollars did they each have originally? Analyze Find the original amounts Formulate Represent the problem as an equation Determine x – 5 = 2(x – 20) x – 5 = 2x – 40 –x – 5 = –40 –x = –35 x = 35 Justify They each began with $35 Evaluate 35 – 5 is double 35 – 20 End Part 1 Mark and Susan are given the same amount of money. Mark spends $5, and Susan spends $20. If Mark now has twice as much money as Susan, how many dollars did they each have originally? First, analyze the problem. The problem presents a word problem about Mark and Susan, who begin with equal amounts of money. When Mark spends $5 and Susan spends $20, Mark has twice as much as Susan. The problem asks for the original amount that they each had. Next, formulate a plan or strategy to solve the problem. Represent the problem as an equation. Simplify each side, eliminate the variable from one side and the constant from the other side, and divide both sides by the coefficient of the variable. Next, determine the solution to the problem. Represent the problem as an equation. Use the distributive property. Subtract 2x from each side. Add 5 to each side. Divide each side by –1. Now that the solution has been determined, justify it. An equation was written showing Mark’s amount (on the left) equal to twice Susan’s amount (on the right). The right side of the equation was simplified, and then the variable term was eliminated from the right, and the constant term was eliminated from the left. Dividing each side by the coefficient of x gave the solution, which shows that Mark and Susan each began with $35. Last, evaluate the effectiveness of the steps, and the reasonableness of the solution. The steps for solving a multistep equation worked effectively. The answer is reasonable because 35 – 5 is double 35 – 20.