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Australia
Year 8

5.03 Distributing to solve

Lesson

Introduction

Algebra is useful when solving equations because it allows for multiple different methods. Depending on the question and the numbers involved, we can choose which is the most appropriate approach, and hopefully save some time.

One of these methods is distributing to solve, and it can save a lot of work when used appropriately.

Distribute in an equation

Using the  distributive law  in an equation with algebraic terms follows the same rules that we use to distribute a numeric expression.

The distributive law allows us to expand multiplication applied to an expression inside a pair of brackets according to the rule:A(B+C)=AB+AC

Using the distributive law when solving an equation can help us reduce the equation to only the four basic operations, since we can use it to expand brackets.

By expanding the brackets first, the order in which we apply reverse operations becomes more obvious.

Let's have a look at what happens if we solve the equation 7(r-2)+5=37 using only reverse operations.

If we do this, we get the working out:

\displaystyle 7(r-2)+5\displaystyle =\displaystyle 37
\displaystyle 7(r-2)\displaystyle =\displaystyle 32Reverse the addition
\displaystyle r-2\displaystyle =\displaystyle \dfrac{32}{7}Reverse the multiplication
\displaystyle r\displaystyle =\displaystyle \dfrac{32}{7}+2Reverse the subtraction
\displaystyle r\displaystyle =\displaystyle \dfrac{32}{7}+\dfrac{14}{7}Convert 2 to a fraction with denominator 7
\displaystyle r\displaystyle =\displaystyle \dfrac{46}{7}Perform the addition

We find that it takes more steps to solve the equation using only reverse operations and it involves a lot more fractions.

As such, while both methods for solving the equation work just fine, using the distributive law can save us some time.

The other reason to use the distributive law when solving equations is that there are some problems we simply can't solve using just reverse operations. Consider the equation:3(p-2) + 4(2p+8)=81

Because we can't reverse the multiplications of 3 and 4 at the same time, it's actually impossible to solve this equation using only reverse operations. But we can solve it using the distributive law.

If we use the distributive law to expand all the brackets first, we will only have multiplication, addition and subtraction left. We can then gather any like terms and apply reverse operations, like so:

\displaystyle 3(p-2) + 4(2p+8)\displaystyle =\displaystyle 81
\displaystyle 3\times p + 3\times2 + 4\times2p + 4\times8\displaystyle =\displaystyle 81Use the distributive law
\displaystyle 3p-6 + 8p+32\displaystyle =\displaystyle 81Simplify the products
\displaystyle 11p+26\displaystyle =\displaystyle 81Combine like terms
\displaystyle 11p\displaystyle =\displaystyle 55Reverse the addition
\displaystyle p\displaystyle =\displaystyle 5Reverse the multiplication

Examples

Example 1

Solve this equation for x by first expanding the brackets:-6(4x+5)=-143

Worked Solution
Create a strategy

Use the distributive law, then reverse the operations.

Apply the idea
\displaystyle -6(4x+5)\displaystyle =\displaystyle -143Write the equation
\displaystyle -6\times 4x + (-6) \times 5\displaystyle =\displaystyle -143Use the distributive law
\displaystyle -24x-30\displaystyle =\displaystyle -143Simplify the products
\displaystyle -24x-30+30\displaystyle =\displaystyle -143+30Add 30 to both sides
\displaystyle -24x\displaystyle =\displaystyle -113Evaluate
\displaystyle \dfrac{-24x}{-24}\displaystyle =\displaystyle \dfrac{-113}{-24}Divide both sides by -24
\displaystyle x\displaystyle =\displaystyle \dfrac{113}{24}Evaluate

Example 2

Solve this equation for x by first expanding the brackets and collecting like terms:4(4x-6)-3x+8=36

Worked Solution
Create a strategy

Use the distributive law, then reverse the operations.

Apply the idea
\displaystyle 4(4x-6)-3x+8\displaystyle =\displaystyle 36Write the equation
\displaystyle 4\times 4x + 4\times (-6) -3x+8\displaystyle =\displaystyle 36Use the distributive law
\displaystyle 16x-24-3x+8\displaystyle =\displaystyle 36Simplify the products
\displaystyle 13x-16\displaystyle =\displaystyle 36Collect like terms
\displaystyle 13x-16+16\displaystyle =\displaystyle 36+16Add 16 to both sides
\displaystyle 13x\displaystyle =\displaystyle 52Evaluate
\displaystyle \dfrac{13x}{13}\displaystyle =\displaystyle \dfrac{52}{13}Divide both sides by 13
\displaystyle x\displaystyle =\displaystyle 4Evaluate

Example 3

Solve this equation for x by first expanding the brackets:-3(2x-6) + 5(4x-5)=21

Worked Solution
Create a strategy

Use the distributive law, then reverse the operations.

Apply the idea
\displaystyle -3(2x-6) + 5(4x-5)\displaystyle =\displaystyle 21Write the equation
\displaystyle -3\times2x + (-3)\times(-6) + 5\times4x + 5\times(-5)\displaystyle =\displaystyle 21Use the distributive law
\displaystyle -6x+18 + 20x-25\displaystyle =\displaystyle 21Simplify the products
\displaystyle 14x-7\displaystyle =\displaystyle 21Collect like terms
\displaystyle 14x-7+7\displaystyle =\displaystyle 21+7Add 7 to both sides
\displaystyle 14x\displaystyle =\displaystyle 28Evaluate
\displaystyle \dfrac{14x}{14}\displaystyle =\displaystyle \dfrac{28}{14}Divide both sides by 14
\displaystyle x\displaystyle =\displaystyle 2Evaluate
Idea summary

The distributive law allows us to expand multiplication applied to an expression inside a pair of brackets according to the rule:A(B+C)=AB+AC

By expanding the brackets first, the order in which we apply reverse operations becomes more obvious and it can save time in solving the equation.

Outcomes

ACMNA194

Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution

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