Solve Linear Equations

An equation is a numerical statement that has an ‘=’ sign.

Example:

    \[x+5=16\]

One of the numbers has been written as a letter. We solve an equation when we calculate the value of the letter.

Since we know that 11+5=16 we can say that x=11. The number 11 is the solution to the equation above.

Equations come in many forms.

Linear equations are equations that we solve by adding, subtracting, multiplying or dividing only.

Vocabulary

Examples
term 3x;   1;     -5x;     -5;   3(2x-1);    \frac{2x-5}{3}
expression  3x+1;     \frac{2x-5}{3}-5x
equation  3x+1=22;    \frac{2x-5}{3}=9
relation  y=3x+1

In an introduction to algebra an equation might look like this:

    \[3 \times \Box + 5 = 26\]

The idea is that you’re supposed to find the number that goes in the box to make the calculation correct.

One big idea of algebra is to use a letter to represent the unknown number until you figure it out. That is:

    \[3 \times x + 5 = 26\]

Which we shorten to:

    \[3x+5=26\]

Solve a ‘two step’ problem

An equation such as 3x+5=26 is often referred to as a two step problem. The value x has been multiplied by 3, and then 5 has been added. Two operations. That requires two reverse operations to figure out x. Here are three techniques for figuring out the value x:

  1. Guess and check. This is a good approach if the unknown is a simple whole number.
  2. Work backwards. This is a good approach for two step equations, but not for more complicated equations.
  3. Completing and balancing. This is the what algebra is named for. Its when we “do the same to both sides” to keep the balance. This method works for all kinds of complicated equations and is preferred by people familiar with complicated work.

Mathantics Scroll down to ‘Algebra Basics Part 1’ for a set of four videos on solving equations.

applet link

Solve Equations with letters on both sides

It can be helpful to follow some predetermined steps to solve these.

  • Expand any brackets, simplify each side if necessary;
  • Add or subtract the x term from one side so that there is an x term on one side only;
  • Continue as before, in skill 4.

There could be a better approach than these three steps – it really depends on the equation you’re solving. This is why it is good to keep thinking about the objective (figuring out what x works) and not get too hung up on one method.

applet link

When there are fractions

Thinking about solving linear equations with fractions

A fraction is an alternate way to express division.

    \[\frac{9}{4}=9 \div 4 = 2.25\]

The inverse operation of division is multiplication:

    \[\frac{9}{4}\times 4 =9\div 4 \times 4 =  9\]

Or we could say:

    \[\frac{9}{4}\times 4 =\frac{9}{4} \times \frac{4}{1} = \frac{9}{\cancel{4}} \times \frac{\cancel{4}}{1}= 9\]

Multiplying by the denominator ‘clears’ the denominator. For example, solve

    \[\frac{58-x}{5}=2x+5$\]

In this case, the denominator is 5. Remember to multiply both sides by 5.

    \begin{align*} 5\times \Big(\frac{58-x}{5}\Big)&=5\times \Big(2x+5\Big)\\[10pt] \frac{5(58-x)}{5}&=10x+25\\[10pt]\frac{\cancel{5}(58-x)}{\cancel{5}}&=10x+25\\[10pt] 58-x&=10x+25\\[10pt]58&=11x+25\\[10pt]33&=11x\\[10pt]x&=3\end{align*}

applet link

More!

Check out these online/interactive/printable learning resources.


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