Lesson Video: Multistep Equations Mathematics • 7th Grade

Video Transcript

In this video, we will learn how to solve multistep equations. We will do this using the balancing method and inverse operations. We will begin by defining a multistep equation and recalling how we solve one-step equations. Multistep equations are algebraic equations that require more than one operation such as addition, subtraction, multiplication, or division to solve. It is important to know about the order of operations when solving multistep equations. This order is sometimes referred to as PEMDAS or BIDMAS. The letters M, D, A, and S that appear in both stand for multiplication, division, addition, and subtraction. The P stands for parentheses, whereas the B stands for brackets and the E stands for exponents and the I indices.

We will now recall how we would solve one-step equations. Let’s consider two equations, 𝑥 plus seven is equal to 19 and six 𝑥 is equal to 30. In order to solve these, we need to recall the inverse operations, for example, addition and subtraction and multiplication and division. In order to use the balancing method, we need to do the same to both sides of the equation. In our first example, we need to subtract seven. This is because subtracting seven is the opposite or inverse of adding seven. The left-hand side simplifies to just 𝑥. 19 minus seven is equal to 12. So the right-hand side is 12. The solution to the equation is 𝑥 equals 12. We could check this by substituting the 12 back in to the original equation.

In our second example, six 𝑥 is equal to 30. We need to divide both sides by six. This is because division is the inverse of multiplication. 30 divided by six is equal to five. Therefore, the solution to this one-step equation is 𝑥 equals five. We will now use this knowledge of inverse operations and the balancing method to solve some multistep equations.

I think of a number. I multiply it by three and then subtract five. The answer is 13. What number did I think of?

One way to answer this question is to set up an equation. We can begin by letting the number being unknown, in this case, 𝑥. We are told that we multiply the number by three. This can be written as three 𝑥. We then subtract five, so our expression is three 𝑥 minus five. As the answer is 13, we can turn this expression into an equation by setting it equal to 13. We can now work out the number by solving this equation. We can do this using the balancing method and inverse operations.

The inverse or opposite of subtracting five is adding five. Adding five to both sides of the equation gives us three 𝑥 equals 18 as negative five plus five is zero and 13 plus five is 18. Our final step is to divide three from both sides of this new equation as division is the inverse of multiplication. Three 𝑥 divided by three is equal to 𝑥, and 18 divided by three is equal to six. This means that the number that was initially thought of was six. We can check this answer by multiplying six by three and then subtracting five. As this does indeed give us an answer of 13, we know that the answer is correct.

An alternative method that could’ve been used in this question is using function machines. Once again, we start with the unknown 𝑥. We multiply it by three, subtract five, and end up with the answer 13. We can reverse this by carrying out the inverse operations. The inverse or opposite of subtracting five is adding five. And the inverse of multiplying by three is dividing by three. 13 plus five is equal to 18. Dividing this by three gives us an answer of six. We have once again proved that the original number was six.

The second question that we’ll look at will be a word problem in context.

A water company bills its customers using the rule 𝑐 equals 10 plus four 𝑚, where 𝑐 is the cost in dollars, 𝑚 is the number of cubic meters of water used, and 10 is the standing charge. They produce a bill for 262 dollars. How many cubic meters of water have been used?

We’re given the rule or equation 𝑐 is equal to 10 plus four 𝑚. We’re also told that the cost 𝑐 is equal to 262 dollars. This means that we can rewrite the equation as 262 is equal to 10 plus four 𝑚. This equation can then be solved using the balancing method. We begin by subtracting 10 from both sides of the equation as subtracting 10 is the opposite of adding 10. 262 minus 10 is equal to 252, which is equal to four 𝑚. Our second and final step is to divide both sides of this equation by four. 252 divided by four is equal to 63. And four 𝑚 divided by four is equal to 𝑚.

One way of calculating 252 divided by four would be to use the short division bus stop method. Four does not divide into two, so we carry the two to the tens column. 25 divided by four is equal to six remainder one. So, we carry the one to the units or ones column. Finally, 12 divided by four is equal to three, so 252 divided by four is equal to 63. We can therefore conclude that 63 cubic meters of water have been used.

Solve eight 𝑘 minus three minus two 𝑘 is equal to 21.

In order to solve this equation, we firstly need to group or collect the like terms. Eight 𝑘 minus two 𝑘 is equal to six 𝑘, so our equation becomes six 𝑘 minus three is equal to 21. From this point onwards, we can use our knowledge of inverse operations and the balancing method. We begin by adding three to both sides. This gives us six 𝑘 is equal to 24. Our final step is to divide both sides of the equation by six. Six 𝑘 divided by six is 𝑘, and 24 divided by six is four.

The solution to the equation eight 𝑘 minus three minus two 𝑘 equals 21 is 𝑘 equals four. We could check this answer by substituting four back in to the original equation.

We will now look at two variations of this type of question.

Solve six 𝑥 minus five equals two 𝑥 plus 11.

In this question, we have an unknown 𝑥 on both sides of the equation. In order to solve an equation of this type, we need to get all the 𝑥 terms on one side of the equal sign and all the constants on the other. We can do this by firstly subtracting two 𝑥 from both sides of the equation. We could also add five to both sides at the same time. On the left-hand side, six 𝑥 minus two 𝑥 is equal to four 𝑥. And negative five plus five is equal to zero. On the right-hand side, two 𝑥 minus two 𝑥 is zero, and 11 plus five is equal to 16. The equation simplifies to four 𝑥 equals 16.

Our final step is to divide both sides of this equation by four as dividing by four is the inverse, or opposite, of multiplying by four. The solution to the equation six 𝑥 minus five equals two 𝑥 plus 11 is 𝑥 equals four. We could substitute this value back in to both sides of the equation to check that our answer is correct. On the left-hand side, we would have six multiplied by four minus five. This is equal to 19. On the right-hand side, we would have two multiplied by four plus 11. As this is also equal to 19, we know that our answer is correct.

Solve 12 minus three 𝑥 equals six.

This question has a negative 𝑥 term. One way of solving this would be to make sure that our 𝑥 term was positive. We can do this by adding three 𝑥 to both sides. The left-hand side simplifies to 12, and the right-hand side to six plus three 𝑥. We can then subtract six from both sides of this new equation. 12 minus six is equal to six, so we have six equals three 𝑥. Dividing both sides of this equation by three gives us two equals 𝑥. Whilst this answer is correct, we often rewrite it as 𝑥 equals two, with the unknown on the left-hand side.

The next question that we’ll look at is another practical problem in context.

Find the value of 𝑎 given the following information: 𝑌 is on the line between 𝑋 and 𝑍. 𝑋𝑌 equals 24 centimeters, 𝑌𝑍 equals eight 𝑎 centimeters, and 𝑋𝑍 equals 88 centimeters.

We can begin this question by drawing the line 𝑋𝑍 on which 𝑌 lies. We’re told that 𝑋𝑌 equals 24 centimeters, 𝑌𝑍 equals eight 𝑎 centimeters, and the length of the whole line 𝑋𝑍 is 88 centimeters. The length from 𝑋𝑌 plus the length from 𝑌𝑍 is equal to the total distance from 𝑋 to 𝑍. As all of our measurements are in centimeters, this can be written as an equation. 24 plus eight 𝑎 is equal to 88. We can then solve this equation to calculate the value of 𝑎. We begin by subtracting 24 from both sides. As 88 minus 24 is equal to 64, we have eight 𝑎 equals 64. We can then divide both sides of this new equation by eight. This gives us the value of 𝑎 equal to eight.

We can check this by substituting this value back in. From the line above, we actually know that eight 𝑎 is equal to 64. This means that the length of the line from 𝑌 to 𝑍 is 64 centimeters. As 24 centimeters plus 64 centimeters equals 88 centimeters, our answer is correct.

Our final question in this video involves solving a more complicated multistep equation.

What value of 𝑥 solves 𝑥 plus one over two minus 𝑥 minus one over three is equal to 𝑥?

In order to solve this equation, we firstly need to consider the left-hand side. In order to add or subtract any fractions, we firstly need to make sure the denominators are the same. We do this by finding the lowest common multiple or LCM. In this case, this would be six. We need to multiply the numerator and denominator of the first fraction by three and the second fraction by two. The first fraction becomes three multiplied by 𝑥 plus one over six. The second fraction, which has been subtracted, becomes two multiplied by 𝑥 minus one over six. This is all equal to 𝑥. As the denominators are now the same, we can write the left-hand side as a single fraction.

Our next step is to distribute the parentheses, otherwise known as expanding the brackets. Three multiplied by 𝑥 is three 𝑥. And three multiplied by one is three. Negative two multiplied by 𝑥 is negative two 𝑥. And negative two multiplied by negative one is positive two. Remember, when we multiply a negative by a negative, our answer is positive. The equation simplifies to three 𝑥 plus three minus two 𝑥 plus two over six is equal to 𝑥. We can now simplify the numerator by collecting or grouping like terms. Three 𝑥 minus two 𝑥 is equal to 𝑥, and three plus two is equal to five. We have 𝑥 plus five over six is equal to 𝑥.

We can now solve this equation using the balancing method and our knowledge of inverse operations. We begin by multiplying both sides by six. This gives us 𝑥 plus five is equal to six 𝑥. As there is a higher coefficient of 𝑥 on the right-hand side, we can subtract 𝑥 from both sides. This gives us five is equal to five 𝑥. Finally, we divide both sides of this equation by five. The value that solves the equation is 𝑥 equals one. We could check this by substituting this value back in to the original equation.

We will now summarize the key points from this video. A multistep equation can be solved using the balancing method and our knowledge of inverse operations. For example, addition and subtraction and multiplication and division are inverse operations. If we add, subtract, multiply, or divide both sides of an equation by the same amount, then the equality is still true. A simple multistep equation can be solved by reversing all the operations in the contrary order to calculate the missing value. This means that we reverse the operations in the opposite order to PEMDAS or BIDMAS.