Solving Multi-step Equations
There are some equations that you can solve in your head in an instant. If I asked you to solve `2y = 4`, chances are you wouldn’t need to get out a pencil and paper to calculate that `y = 2`. You only needed to do one thing to get the answer: divide by `2`.
Other equations are more complicated. Can you solve `6(1/4t+3/8)=2` without writing anything down? Me neither. That’s because it contains not just a variableA symbol that represents an unknown value. but also fractions and terms inside parentheses. This is a multi-step equationAn equation that requires more than one step to solve., one that takes several operations to solve. Don’t worry though. Although they take more time and more operations, multi-step equations can still be simplified and solved by applying basic algebraic rules. Two helpful tools are the Distributive PropertyStates that the product of a number and a sum equals the sum of the individual products of the number and the addends: for all real numbers `a`, `b`, and `c`, `a(b + c) = ab + ac`. and multiplication by a common denominatorA number that is a multiple of all of the denominators in a group of fractions..
Just as with simpler equations, solving multi-step equations usually means isolating a variable on one side of the equals sign. Step-by-step, we must get the variable out of parentheses, away from other terms, and with a coefficient of `1`. If a variable is inside parentheses, they can be cleared by applying the Distributive Property of Multiplication.
The Distributive Property states that for all real numbers `a`, `b`, and `c`, `a(b+c)=ab+ac`. What that means is that when a number multiplies an expression inside parentheses, we can distribute the multiplication to each term of the expression individually. Let’s go back to the equation `6(1/4t+3/8)=2`. We can apply the distributive property and clear the parentheses by multiplying each term inside of them by `6`. The expression `6(1/4t+3/8)` then becomes `6/4t+18/8`. Now there are no parentheses.
`6(1/4t+3/8)=2`
`6(1/4t)+6(3/8)=2`
`6/4t+18/8=2`
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In which of the following equations is the distributive property properly applied to the equation `x(y+3)=7`?
A) `y+3x=7`
B) `xy+3x=7x`
C) `xy+3x=7`
D) `xy+3=7`
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We’ve seen how to simplify a multi-step equation by applying the Distributive Property to clear parentheses. It’s also helpful to simplify equations that include fractions before we try to solve them. We can do this by multiplying both sides by a common denominator of all of the fractions in the equation.
Let’s go back to the equation `6/4t+18/8=2`. The lowest common denominator of the fractions `6/4` and `18/8` is `8`. So we’ll multiply the expression with those fractions by `8`. Because we must preserve the equality of the equation, we will also multiply the expression on the other side by `8`. Now we’ve got `8(6/4t+18/8)=8(2)`.
But wait—we just put parentheses back into the equation again. Now what? Simple, we go back to our old friend the Distributive Property once again. Once we multiply each term inside the parentheses by `8`, both the parentheses and the fractions will disappear. Sweet.
`6/4t+18/8=2`
`8(6/4t+18/8)=8(2)`
`8(6/4t)+8(18/8)=8(2)`
`12t+18=16`
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In order to clear away the fractions from `5/3-(2y)/6=19`, we can multiply both sides of the equation by which of the following numbers?
`3` `6` `9` `18`
A) `6`
B) `3` and `6`
C) `9`
D) `6` and `18`
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Complex, multi-step equations often require multi-step solutions. Before we can begin to isolate a variable, we might need to simplify the equation first. This may mean using the Distributive Property to remove parentheses, or multiplying both sides of an equation by a common denominator to get rid of fractions. Sometimes it requires both techniques.