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In this video, we will learn how to solve multistep equations.
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We will do this using the balancing method and inverse operations.
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We will begin by defining a multistep equation and recalling how we solve one-step equations.
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Multistep equations are algebraic equations that require more than one operation such as addition, subtraction, multiplication, or division to solve.
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It is important to know about the order of operations when solving multistep equations.
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This order is sometimes referred to as PEMDAS or BIDMAS.
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The letters M, D, A, and S that appear in both stand for multiplication, division, addition, and subtraction.
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The P stands for parentheses, whereas the B stands for brackets and the E stands for exponents and the I indices.
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We will now recall how we would solve one-step equations.
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Letβs consider two equations, π₯ plus seven is equal to 19 and six π₯ is equal to 30.
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In order to solve these, we need to recall the inverse operations, for example, addition and subtraction and multiplication and division.
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In order to use the balancing method, we need to do the same to both sides of the equation.
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In our first example, we need to subtract seven.
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This is because subtracting seven is the opposite or inverse of adding seven.
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The left-hand side simplifies to just π₯.
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19 minus seven is equal to 12.
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So the right-hand side is 12.
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The solution to the equation is π₯ equals 12.
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We could check this by substituting the 12 back in to the original equation.
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In our second example, six π₯ is equal to 30.
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We need to divide both sides by six.
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This is because division is the inverse of multiplication.
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30 divided by six is equal to five.
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Therefore, the solution to this one-step equation is π₯ equals five.
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We will now use this knowledge of inverse operations and the balancing method to solve some multistep equations.
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I think of a number.
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I multiply it by three and then subtract five.
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The answer is 13.
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What number did I think of?
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One way to answer this question is to set up an equation.
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We can begin by letting the number being unknown, in this case, π₯.
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We are told that we multiply the number by three.
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This can be written as three π₯.
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We then subtract five, so our expression is three π₯ minus five.
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As the answer is 13, we can turn this expression into an equation by setting it equal to 13.
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We can now work out the number by solving this equation.
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We can do this using the balancing method and inverse operations.
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The inverse or opposite of subtracting five is adding five.
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Adding five to both sides of the equation gives us three π₯ equals 18 as negative five plus five is zero and 13 plus five is 18.
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Our final step is to divide three from both sides of this new equation as division is the inverse of multiplication.
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Three π₯ divided by three is equal to π₯, and 18 divided by three is equal to six.
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This means that the number that was initially thought of was six.
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We can check this answer by multiplying six by three and then subtracting five.
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As this does indeed give us an answer of 13, we know that the answer is correct.
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An alternative method that couldβve been used in this question is using function machines.
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Once again, we start with the unknown π₯.
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We multiply it by three, subtract five, and end up with the answer 13.
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We can reverse this by carrying out the inverse operations.
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The inverse or opposite of subtracting five is adding five.
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And the inverse of multiplying by three is dividing by three.
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13 plus five is equal to 18.
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Dividing this by three gives us an answer of six.
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We have once again proved that the original number was six.
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The second question that weβll look at will be a word problem in context.
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A water company bills its customers using the rule π equals 10 plus four π, where π is the cost in dollars, π is the number of cubic meters of water used, and 10 is the standing charge.
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They produce a bill for 262 dollars.
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How many cubic meters of water have been used?
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Weβre given the rule or equation π is equal to 10 plus four π.
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Weβre also told that the cost π is equal to 262 dollars.
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This means that we can rewrite the equation as 262 is equal to 10 plus four π.
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This equation can then be solved using the balancing method.
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We begin by subtracting 10 from both sides of the equation as subtracting 10 is the opposite of adding 10.
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262 minus 10 is equal to 252, which is equal to four π.
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Our second and final step is to divide both sides of this equation by four.
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252 divided by four is equal to 63.
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And four π divided by four is equal to π.
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One way of calculating 252 divided by four would be to use the short division bus stop method.
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Four does not divide into two, so we carry the two to the tens column.
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25 divided by four is equal to six remainder one.
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So, we carry the one to the units or ones column.
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Finally, 12 divided by four is equal to three, so 252 divided by four is equal to 63.
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We can therefore conclude that 63 cubic meters of water have been used.
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Solve eight π minus three minus two π is equal to 21.
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In order to solve this equation, we firstly need to group or collect the like terms.
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Eight π minus two π is equal to six π, so our equation becomes six π minus three is equal to 21.
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From this point onwards, we can use our knowledge of inverse operations and the balancing method.
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We begin by adding three to both sides.
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This gives us six π is equal to 24.
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Our final step is to divide both sides of the equation by six.
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Six π divided by six is π, and 24 divided by six is four.
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The solution to the equation eight π minus three minus two π equals 21 is π equals four.
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We could check this answer by substituting four back in to the original equation.
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We will now look at two variations of this type of question.
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Solve six π₯ minus five equals two π₯ plus 11.
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In this question, we have an unknown π₯ on both sides of the equation.
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In order to solve an equation of this type, we need to get all the π₯ terms on one side of the equal sign and all the constants on the other.
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We can do this by firstly subtracting two π₯ from both sides of the equation.
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We could also add five to both sides at the same time.
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On the left-hand side, six π₯ minus two π₯ is equal to four π₯.
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And negative five plus five is equal to zero.
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On the right-hand side, two π₯ minus two π₯ is zero, and 11 plus five is equal to 16.
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The equation simplifies to four π₯ equals 16.
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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.
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The solution to the equation six π₯ minus five equals two π₯ plus 11 is π₯ equals four.
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We could substitute this value back in to both sides of the equation to check that our answer is correct.
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On the left-hand side, we would have six multiplied by four minus five.
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This is equal to 19.
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On the right-hand side, we would have two multiplied by four plus 11.
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As this is also equal to 19, we know that our answer is correct.
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Solve 12 minus three π₯ equals six.
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This question has a negative π₯ term.
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One way of solving this would be to make sure that our π₯ term was positive.
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We can do this by adding three π₯ to both sides.
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The left-hand side simplifies to 12, and the right-hand side to six plus three π₯.
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We can then subtract six from both sides of this new equation.
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12 minus six is equal to six, so we have six equals three π₯.
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Dividing both sides of this equation by three gives us two equals π₯.
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Whilst this answer is correct, we often rewrite it as π₯ equals two, with the unknown on the left-hand side.
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The next question that weβll look at is another practical problem in context.
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Find the value of π given the following information: π¦ is on the line between π₯ and π§.
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π₯π¦ equals 24 centimeters, π¦π§ equals eight π centimeters, and π₯π§ equals 88 centimeters.
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We can begin this question by drawing the line π₯π§ on which π¦ lies.
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Weβre told that π₯π¦ equals 24 centimeters, π¦π§ equals eight π centimeters, and the length of the whole line π₯π§ is 88 centimeters.
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The length from π₯π¦ plus the length from π¦π§ is equal to the total distance from π₯ to π§.
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As all of our measurements are in centimeters, this can be written as an equation.
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24 plus eight π is equal to 88.
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We can then solve this equation to calculate the value of π.
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We begin by subtracting 24 from both sides.
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As 88 minus 24 is equal to 64, we have eight π equals 64.
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We can then divide both sides of this new equation by eight.
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This gives us the value of π equal to eight.
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We can check this by substituting this value back in.
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From the line above, we actually know that eight π is equal to 64.
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This means that the length of the line from π¦ to π§ is 64 centimeters.
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As 24 centimeters plus 64 centimeters equals 88 centimeters, our answer is correct.
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Our final question in this video involves solving a more complicated multistep equation.
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What value of π₯ solves π₯ plus one over two minus π₯ minus one over three is equal to π₯?
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In order to solve this equation, we firstly need to consider the left-hand side.
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In order to add or subtract any fractions, we firstly need to make sure the denominators are the same.
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We do this by finding the lowest common multiple or LCM.
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In this case, this would be six.
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We need to multiply the numerator and denominator of the first fraction by three and the second fraction by two.
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The first fraction becomes three multiplied by π₯ plus one over six.
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The second fraction, which has been subtracted, becomes two multiplied by π₯ minus one over six.
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This is all equal to π₯.
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As the denominators are now the same, we can write the left-hand side as a single fraction.
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Our next step is to distribute the parentheses, otherwise known as expanding the brackets.
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Three multiplied by π₯ is three π₯.
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And three multiplied by one is three.
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Negative two multiplied by π₯ is negative two π₯.
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And negative two multiplied by negative one is positive two.
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Remember, when we multiply a negative by a negative, our answer is positive.
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The equation simplifies to three π₯ plus three minus two π₯ plus two over six is equal to π₯.
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We can now simplify the numerator by collecting or grouping like terms.
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Three π₯ minus two π₯ is equal to π₯, and three plus two is equal to five.
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We have π₯ plus five over six is equal to π₯.
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We can now solve this equation using the balancing method and our knowledge of inverse operations.
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We begin by multiplying both sides by six.
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This gives us π₯ plus five is equal to six π₯.
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As there is a higher coefficient of π₯ on the right-hand side, we can subtract π₯ from both sides.
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This gives us five is equal to five π₯.
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Finally, we divide both sides of this equation by five.
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The value that solves the equation is π₯ equals one.
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We could check this by substituting this value back in to the original equation.
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We will now summarize the key points from this video.
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A multistep equation can be solved using the balancing method and our knowledge of inverse operations.
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For example, addition and subtraction and multiplication and division are inverse operations.
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If we add, subtract, multiply, or divide both sides of an equation by the same amount, then the equality is still true.
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A simple multistep equation can be solved by reversing all the operations in the contrary order to calculate the missing value.
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This means that we reverse the operations in the opposite order to PEMDAS or BIDMAS.