02Gequations

= Solving Equations = toc Recall, an ** equation **, has one "=" and an expression on each side of the equals sign.

To ** solve an equation ** means to find a value for the pronumeral that makes the equation true.

For example, with the equation **//x + 3 = 7//**


 * // **x = 4** // __ is __ a solution because
 * when // **x = 4** // is substituted into the equation
 * // **x + 3** // __ does __ equal // **7** //.


 * // **x = 1** // is __ not __ a solution because
 * when // **x = 1** // is substituted into the equation
 * // **x + 3** // does __ not __ equal // **7** //.

Linear Equations
A ** linear equation ** has:
 * only one variable (pronumeral) {though it can appear more than once}
 * the variable is not raised to any power {eg not squared or cubed, etc}
 * the variable is not inside a root sign {eg square root or cube root etc}

The following are linear equations: math . \quad \centerdot \; \dfrac{x+3}{5} = 9 math
 * 3x + 4 = 10
 * 5(y – 2) = 4y + 9
 * m = 7

The following are __not__ linear equations math \\ . \quad \centerdot \; \sqrt{x - 5} = 7 \\ \\ . \quad \centerdot \; x^2 + 5x - 7 = 9 \\ \\ . \quad \centerdot \; 3x + y = 9 math

Solving 1-Step Equations
To solve the equation, you must undo what has been done to the variable. This means applying the reverse operation to each side of the equation.

It is important that we do exactly the same thing to both sides of the equation so that the overall value of the equation is not changed.

Note: The solution to one step equations are often obvious to most people. It is important to learn the way to set out the solution clearly so that you have the skills to tackle more advanced equations.

** Example 1 **
math \textbf{(a)} \quad \text{Solve : } x + 3= 7 math Note: Notice the way the equals signs have been lined up underneath each other.

This way of setting out can be abbreviated by reducing the middle line -- like this:

** Example 2 **
math \textbf{(a)} \quad \text{Solve : } x - 8 = 10 math

math \textbf{(b)} \quad \text{Solve : } 4x = 20 math math \textbf{(c)} \quad \text{Solve : } \dfrac{x}{7} = 3 math

Solving Multi-Step Equations
To solve multi-step equations, we undo what has been done to the variable in __reverse order of operations__.

This means, decide the __last__ thing that was done to the variable and undo that.

** Example 3 **
math \textbf{(a)} \quad \text{Solve : } 2x-5=9 math

{x has been __multiplied by 2__, THEN __subtracted 5__ so undo the __last__ thing}

{The opposite of __subtract 5__ is __add 5__}

math \textbf{(b)} \quad \text{Solve : } \dfrac{x}{3} + 1=6 math

math \textbf{(c)} \quad \text{Solve : } \dfrac{x-3}{4}=-2 math



math \textbf{(d)} \quad \text{Solve : } 2 \big( x-5 \big) = 1 math

Where the variable is on both sides
If the variable appears on both sides of the equals signs,
 * expand any brackets in the equation
 * eliminate the term containing the variable from one side by either adding or subtracting that term
 * then solve like a normal multi-step equation

Hint : Eliminate the term with the lower value {reduces the amount of negatives you have to deal with}

** Example 4 **
math \textbf{(a)} \quad \text{Solve : } 7x-5=3x+9 math



math \textbf{(b)} \quad \text{Solve : } 2 \big( x-5 \big) = -3x-7 math

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