Equations

NZ Level 6 (NZC) Level 1 (NCEA)

Compare linear relationships

Lesson

We've already learnt how to identify linear equations, which are shown graphically as straight line graphs. Now we are going to learn how to compare the features of linear equations. Linear equations are often written in gradient-intercept form which is handy because it helps us identify the gradient and $y$`y`-intercepts of these lines as shown below.

- Gradient: we may be asked which of the two linear equations has a greater or lesser gradient. This may also be referred to as the steepness of the line, the rate of change or simply increasing or decreasing faster/slower. The gradient refers to how much the dependent variable changes for every time the independent variable increases by one.
- $y$
`y`-intercept: the $y$`y`-intercept is where the line crosses the $y$`y`axis. The line with the greater $y$`y`-intercept will be the one that crosses at a higher number on the $y$`y`axis. - $x$
`x`-intercept: this is where the line crosses the $x$`x`axis. Just like the y-intercept, the line with the greater $x$`x`-intercept will be the one that crosses at a higher number on the $x$`x`axis. - Coordinates: each pair of coordinates has an $x$
`x`and a $y$`y`value $\left(x,y\right)$(`x`,`y`). We may be asked to substitute an $x$`x`or a $y$`y`value into a pair of linear equations to see which equation produces the greater/ smaller value for the other variable.

Let's look through some examples that compare these different features of linear equations.

In which of the following is $y$`y` increasing faster?

$x$ `x`$0$0 $1$1 $2$2 $y$ `y`$3$3 $10$10 $17$17 ALoading Graph...B$x$ `x`$0$0 $1$1 $2$2 $y$ `y`$3$3 $10$10 $17$17 ALoading Graph...B

Which of the following has the higher $y$`y`-intercept?

The line with a gradient of $4$4 that crosses the $y$

`y`-axis at $\left(0,6\right)$(0,6).AThe line given by the equation $y=x+4$

`y`=`x`+4BThe line with a gradient of $4$4 that crosses the $y$

`y`-axis at $\left(0,6\right)$(0,6).AThe line given by the equation $y=x+4$

`y`=`x`+4B

For both linear relationships, consider when $y$`y` has a value of $46$46. Which has the smaller corresponding $x$`x` value?

$y=2x+32$

`y`=2`x`+32A$x$ `x`$0$0 $1$1 $2$2 $y$ `y`$6$6 $10$10 $14$14 B$y=2x+32$

`y`=2`x`+32A$x$ `x`$0$0 $1$1 $2$2 $y$ `y`$6$6 $10$10 $14$14 B

Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns

Apply algebraic procedures in solving problems