Coaching Centre
Equations and word problems
Solving linear equations with brackets, fractions and the unknown on both sides, solving after substitution, and turning a worded problem into an equation you can solve and check.
An equation is a balance. Whatever you do to one side, you do to the other, and the balance stays level. Solving means doing the right things in the right order until the pronumeral is alone.
Keeping the balance
Each step undoes something that was done to $x$. Addition is undone by subtraction, multiplication by division, and the other way round. If $x$ appears on both sides, first move all the $x$ terms to one side.
Solve $5(x - 3) = 2x + 9$.
Solve $3x + 5 = 20$ by typing one operation at a time, such as -5 then /3. Then try $5x - 4 = 2x + 11$. You can use an $x$ term, for example -2x. Can you solve it in exactly three moves?
Equations with fractions
Fractions are the main source of slips. The cleanest method is to multiply both sides by the denominator straight away, so the fractions disappear.
Solve $\dfrac{2x + 1}{3} = x - 4$.
Solving after substitution
Often the formula is given and the unknown isn't the subject. You don't need to rearrange first. Substitute everything you know, then solve the equation that's left.
Use $v = u + at$ to find the acceleration $a$ when $v = 30$, $u = 6$ and $t = 8$.
From words to an equation
Worded problems are where Standard marks are won. Use the same four moves every time:
- Let a pronumeral stand for the thing you're asked to find, with units.
- Write an equation that says the same thing as the words.
- Solve it.
- Answer the question in a sentence, and check the answer makes sense.
A gym charges a $\$49$ joining fee plus $\$14.50$ a week. Priya has paid $\$455$ in total. How many weeks has she been a member?
An adult ticket to the Parramatta Lanes festival costs $\$12$ more than a child ticket. Two adults and three children pay $\$214$. Find the price of a child ticket.
Your turn
A plumber charges a $\$90$ call-out fee plus $\$75$ an hour. A bill came to $\$352.50$. How long did the job take?
Show the working
Let $h$ be the number of hours. $90 + 75h = 352.5$, so $75h = 262.5$ and $h = 3.5$. The job took $3.5$ hours, which is $3$ hours $30$ minutes.
Solve $4x - 7 = 21$.
$4x = 28$, so $x = 7$.
Solve $3(2x + 5) = 51$.
Expand: $6x + 15 = 51$. Then $6x = 36$ and $x = 6$.
Solve $7x - 4 = 3x + 18$.
Subtract $3x$: $4x - 4 = 18$. Add $4$: $4x = 22$. So $x = 5.5$.
Solve $\dfrac{x}{5} + 2 = 9$.
Subtract $2$: $\dfrac{x}{5} = 7$. Multiply by $5$: $x = 35$.
Solve $\dfrac{3x - 1}{4} = 5$.
Multiply by $4$: $3x - 1 = 20$. So $3x = 21$ and $x = 7$.
"Six more than three times a number is $39$." Which equation matches?
Three times the number is $3n$. Six more than that is $3n + 6$.
Use $v = u + at$ to find $a$ when $v = 30$, $u = 6$ and $t = 8$.
$30 = 6 + 8a$, so $8a = 24$ and $a = 3$.
A taxi charges $\$4.20$ flagfall plus $\$2.30$ per kilometre. A fare was $\$50.20$. How many kilometres was the trip?
$4.2 + 2.3k = 50.2$, so $2.3k = 46$ and $k = 20$ km.
Exam style: A rectangle's length is $5$ cm more than twice its width. Its perimeter is $64$ cm. Find the width in cm.
Let the width be $w$. The length is $2w + 5$. Perimeter: $2(2w + 5) + 2w = 64$, so $6w + 10 = 64$, $6w = 54$ and $w = 9$ cm.
Exam style: Four friends share a bill. Three pay the same amount and the fourth pays $\$12$ less. The bill is $\$140$. How much does each of the three pay?
Let each of the three pay $p$. Then $3p + (p - 12) = 140$, so $4p = 152$ and $p = 38$. Each pays $\$38$ and the fourth pays $\$26$.