01Alineareqns

= Solving Equations =

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 radical sign {eg square root or cube root etc}

Solving Equations

 * To solve a linear equation, 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.


 * We undo an operation by doing the reverse.


 * But we have to do the same thing to BOTH sides of the equation.

** Example 1 **
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 {this reduces the amount of negatives you have to deal with}

** Example 2 **
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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