Equations

Lesson

When we want to solve linear equations, we are basically trying to find an unknown value.

We built up our knowledge of solving equations over the last few chapters. Firstly, we talked about the importance of keeping equations equivalent or balanced. Basically this meant that whatever operation we did to one side of the equation, we had to do to the other.

Then we looked at how to solve one, two and three step equations by using the inverse order of operations.

In some questions, we have algebraic terms on both sides of the equation. To solve these, we need to get all the algebraic terms on one side and all the numbers on the other. Then we can solve the equation as usual.

Solve the following equation: $2\left(2x+5\right)=3\left(x+5\right)$2(2`x`+5)=3(`x`+5).

**Think: **In order to get $x$`x` on its own, we will need to expand the brackets and then collect the $x$`x` terms on one side of the equation.

**Do: **First we are going to expand the brackets:

$4x+10=3x+15$4`x`+10=3`x`+15

Then we are going to move the algebraic terms to one side and numbers to the other using subtraction:

$4x+10$4x+10 |
$=$= | $3x+15$3x+15 |
(subtract $3x$3x from both sides) |

$x+10$x+10 |
$=$= | $15$15 | (subtract $10$10 from both sides) |

$x$x |
$=$= | $5$5 |

Once you are able to group variables and solve equations, you can use these skills to solve word problems.

Sally and Eileen do some fundraising for their sports team. Together they raised $\$600$$600. If Sally raised $\$272$$272 more than Eileen, and Eileen raised $\$p$$`p`:

**a)** Write an equation in terms of $p$`p` that represents the relationship between the different amounts and solve for $p$`p`.

**Think: **How can we write this information algebraically?

If Eileen raised $\$p$$`p`, then the amount that Sally raised would be $\$p+272$$`p`+272.

**Do: **We know the total amount fundraised, and also know the expression for how much each person fundraised. We can write this as an equation.

$p+p+272$p+p+272 |
$=$= | $600$600 |

$2p+272$2p+272 |
$=$= | $600$600 |

$2p$2p |
$=$= | $328$328 |

$p$p |
$=$= | $164$164 |

In other words, Eileen raised $\$164$$164.

**b) **Then calculate how much Sally raised.

**Think**: We can substitute in the value of $p$`p` that we found in part (a) to find out how much Sally raised.

**Do:**

$p+272$p+272 |
$=$= | $164+272$164+272 |

$=$= | $\$436$$436 |

Sally raised $\$436$$436.

**Check:** $436+164=\$600$436+164=$600

Solve the following equation:

$4x+24=x+15$4`x`+24=`x`+15

Solve the following equation for $r$`r`:

$12r+8=116-6r$12`r`+8=116−6`r`

Solve the following equation: $2\left(2x+5\right)=3\left(x+5\right)$2(2`x`+5)=3(`x`+5)

Solve first-degree equations involving one variable, including equations with fractional coefficients