Problem solving with equations II

Lesson

Practice

Lesson

We've already looked at how to generate and solve equations in Solving Life's Problems with Equations. In this chapter we're going to look at some more examples, including how to graph equations.

The equations we generate usually have more than one variable (e.g. $x$x and $y$y). We can change the subject of the equations and substitute in values to find solutions.

Let's look at some examples now.

Worked Examples

Question 1

Fahrenheit and Celsius are two different units used to measure temperature. To convert a temperature measure in Celsius to a temperature measure in Fahrenheit, we need to multiply the Celsius measure by $\frac{9}{5}$95 and add $32$32 to the result.

Let $F$F be the temperature measure in Fahrenheit and $C$C be the temperature measure in Celsius. Form an equation relating $F$F and $C$C, with $F$F as the subject of the formula.
Using the conversion formula $F=32+\frac{9C}{5}$F=32+9C5, where $F$F is the temperature in Fahrenheit and $C$C is the temperature in Celcius, find $C$C if $F=122$F=122.

Question 2

A racing car starts the race with $120$120 litres of fuel. From there, it uses fuel at a rate of $3$3 litres per minute.

Complete the table of values:

Number of minutes passed ($x$x) $0$0 $5$5 $10$10 $15$15 $20$20 $40$40

Amount of fuel left in tank ($y$y) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Write an algebraic relationship linking the number of minutes passed ($x$x) and the amount of fuel left in the tank ($y$y).
Using the table of values, graph the amount of fuel remaining ($y$y) after $x$x minutes.

Loading Graph...

Number of minutes passed ($x$`x`)	$0$0	$5$5	$10$10	$15$15	$20$20	$40$40
Amount of fuel left in tank ($y$`y`)	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

Question 3

A $carpenter$ charges a callout fee of $\$90$$90 plus $\$30$$30 per hour.

Write an equation to represent the total amount charged $y$y by the $carpenter$ as a function of the number of hours worked $x$x.
Complete the table of values:

Number of hours worked ($x$x) $0$0 $1$1 $2$2 $3$3 $4$4 $4.5$4.5 $7$7

Total fee charged ($y$y) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Graph the relationship between the number of hours worked ($x$x) and the total fee charged ($y$y).

Loading Graph...

Number of hours worked ($x$`x`)	$0$0	$1$1	$2$2	$3$3	$4$4	$4.5$4.5	$7$7
Total fee charged ($y$`y`)	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

Question 4

Matches were used to make the pattern attached:

Complete the table:

Number of triangles ($t$t) $1$1 $2$2 $3$3 $5$5 $10$10 $20$20

Number of matches ($m$m) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Write a formula that describes the relationship between the number of matches ($m$m) and the number of triangles ($t$t).

$m$m = $\editable{}$ $t$t
How many matches are required to make $25$25 triangles using this pattern?

Number of triangles ($t$`t`)	$1$1	$2$2	$3$3	$5$5	$10$10	$20$20
Number of matches ($m$`m`)	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

Outcomes

9.A.P.1

Definition of a polynomial in one variable, its coefficients, with examples and counterexamples, its terms, zero polynomial. Degree of a polynomial. Constant, linear, quadratic, cubic polynomials; monomials, binomials, trinomials. Factors and multiples. Zeros/roots of a polynomial/equation.

9.A.LETV.1

Recall of linear equations in one variable. Introduction to the equation in two variables. Prove that a linear equation in two variables has infinitely many solutions, and justify their being written as ordered pairs of real numbers, plotting them and showing that they seem to lie on a line.