After learning the definitions and concepts of proportional and inversely proportional in mathematics, the next step to study is linear functions. Similar to the graph of direct proportionality, it is linear functions that if the value of $x$ increases, the value of $y$ increases in the same ratio.

In linear functions, there are two concepts: slope and intercept. The shape of the graph will change depending on what number of slopes and intercepts are used. Linear functions also require us to write graphs and create linear equations from them.

There are also many problems to find the intersection, in which case we can use simultaneous equations to find the intersection point.

Linear functions are almost identical to the concept of direct proportionality. Therefore, we frequently use linear functions in our daily life. We will explain how to solve problems with linear functions.

Table of Contents

- 1 Linear functions Are Special Proportional
- 2 How to Write a Linear Function Graph
- 3 Find the Equation for a Linear Function from Intercept and Coordinates
- 4 How to Find the Intersection of Coordinates Using Linear Functions
- 5 Exercises: Linear Function Graphs and Word Problems
- 6 Create and Solve an Equation for Linear Functions

## Linear functions Are Special Proportional

There is often a relationship between direct proportionality. Therefore, we frequently use proportional in everyday life. Another concept that is also essential in everyday life is linear functions. Proportional and linear functions are similar, and all people use linear functions in everyday life.

For example, we use linear functions when we think about our daily living expenses. It is the calculation of a linear function to figure out how much you can spend each day based on your current savings.

The proportional formula is $y=ax$. For example, if you get an allowance of \$20 per month and you don’t spend any money at all, your savings after $x$ months will be expressed by the following equation.

- $y=20x$

If you want to save money from zero savings, it becomes a linear equation like this On the other hand, if you already have some savings, you have to take into account the amount of savings.

For example, if you already have \$50 in savings, the total amount of savings after $x$ months will be following.

- $y=20x+50$

In real life, it is common to start from a specific point, rather than starting from zero. In linear functions, we create an equation that takes this into account.

Think of linear functions as a special case of proportional. Linear functions are proportional that start from a specific point, rather than starting from zero.

### Linear Functions Have a Slope and Intercept, Resulting in the Formula $y=ax+b$

The formula for direct proportionality is $y=ax$. The formula for linear functions, on the other hand, is $y=ax+b$. Because it starts at a specific point, we add $+b$ to the formula of proportional.

$b$ is the value for $x=0$. In the previous example, you already had \$50 in savings. Therefore, $b=50$.

In linear functions, $a$ of $y=ax+b$ is called the slope. Also, $b$ is called the intercept. Remember these words because linear function problems frequently ask for the numbers of the slope and the intercept.

For example, suppose we have the equation $y=-2x-5$. In this case, the slope is -2. Also, the intercept is -5. The concept is not difficult, as it is almost the same formula as the proportional. Just make sure you learn the words.

**-Some Problems Have A Domain**

Mathematics is studied for its application in everyday life. Therefore, some math problems have a domain of variability.

For example, suppose you want to fill a family pool with water. However, the amount of water that can go into the pool is fixed. You can’t pour water into the pool indefinitely.

Therefore, the range of $x$ may be determined. This is called a domain. Also, because of the $x$ domain, the range of $y$ can be determined.

## How to Write a Linear Function Graph

When learning linear functions, we must always understand how to write a graph at the same time. How should we write a graph in a linear function?

Linear functions have a property. It is that the graph of $y=ax+b$ always passes through the coordinates of $(0,b)$. For example, let’s substitute $x=0$. We get the following.

$y=a×0+b$

$y=b$

Thus, if $x=0$, then $y=b$. Since it passes through the coordinates of $(0,b)$, the coordinate of intersection with the $y$ axis is always $b$.

So, when writing a graph of a linear function, focus on the intercept first. For example, the graph of $y=-2x+3$ will be a graph passing through the coordinates of $(0,3)$.

Next, find the $y$ coordinate by putting your favorite number in $x$. For example, in the graph of $y=-2x+3$, if $x=1$, then $y=1$. That is, it passes through the coordinates of $(1,1)$. Then connect the two points $(0,3)$ and $(1,1)$ with a straight line.

The graph of $y=-2x+3$ is a line passing through two points $(0,3)$ and $(1,1)$. You can substitute any number for $x$: 1, 2, 3, etc. to get the coordinates. It is preferable to use numbers that are easy to calculate.

Note that linear function is a linear graph of direct proportionality. Since it is always a linear graph, we can write a graph of a linear function by connecting the two coordinates. How to write a graph can be summarized as follows.

- Find the intercept, $(0,b)$.
- Substitute a number for $x$ and get arbitrary coordinates
- Connect the two coordinates in a straight line

This order will allow us to write a linear function line on the graph.

**-The Greater the Slope, the Steeper It Becomes**

How much the $y$ coordinate increases or decreases depends greatly on the slope. The larger the value of the slope $a$, the steeper the graph will naturally be. Also if the value of the slope $a$ is small, the slope of the graph will be slower.

Also, if the slope is a positive number, the graph will be a straight line rising to the right. On the other hand, if the slope is a negative number, the graph will be a downward straight line to the right.

Make sure you understand this property in a linear function graph.

### Rate of Change: How to Write a Graph Including Fractions

In linear functions, we may be asked about the rate of change. What is the rate of change? Understand that the rate of change is the slope of the linear function. The rate of change and the slope have the same meaning. So when you are asked about the rate of change in a linear function problem, try to answer the slope.

The rate of change is the amount that the value of $y$ changes as the number of $x$ increases. For example, for a linear function with $y=2x+3$, when $x$ increases from 0 to 2, $y$ increases from 3 to 7. When $x$ increases by 2, $y$ increases by 4, so the rate of change (slope) is 2.

We can get the slope from the amount of change in the number of $x$ and $y$. So even if we don’t know the slope of a linear function, we can calculate the slope (rate of change) from a table or graph.

The rate of change can be expressed by the following formula.

Looking at the formula alone, it seems difficult. It also makes us wonder if we have to remember this formula. However, the rate of change in a linear function has the same meaning as the slope. We don’t need to remember the formula because we can just get the slope.

For example, what equation would the following graph?

Many people have trouble about writing linear functions with fractions and finding the equation. But even if the slope of a linear function is a fraction, we can still solve the problem if we understand the rate of change.

If we focus on the $y$ axis, we can find the intercept. So, in the graph above, $b=2$. Next, let’s calculate the rate of change in order to get the slope.

If we check the coordinates of the graph, we can find specific numbers at the following coordinates.

- $(4,5)$
- $(0,2)$
- $(-4,-1)$

If we check these coordinates, we can see that the value of $x$ is increased by 4 and the value of $y$ is increased by 3. The result is as follows.

Therefore, the rate of change is $\displaystyle\frac{3}{4}$. We do not need to remember the formula to produce the rate of change, we just need to produce the slope to satisfy the condition of direct proportionality.

The formula for a proportional is $y=ax$. As the value of $x$ increases by 4, the value of $y$ increases by 3. Substituting the number into the formula, we get $4=3a$. Calculating this gives us $a=\displaystyle\frac{3}{4}$, which is the slope. In other words, the linear function of the graph is the following equation.

- $y=\displaystyle\frac{3}{4}x+2$

Alternatively, we can find the slope by fitting to a linear function formula. From the graph, we already know that $b=2$. Since we know the intercept, the equation in the graph is $y=ax+2$. Let’s assign here the coordinates of $(4,5)$ or $(-4,-1)$.

For example, substituting $(-4,-1)$ would result in the following.

$-1=-4a+2$

$4a=2+1$

$4a=3$

$a=\displaystyle\frac{3}{4}$

In this way, we can get the slope from the formula of linear functions. There is no need to remember the formula for the rate of change. If you try hard to remember many formulas, you will forget them. So try to remember only the most important formulas so that you can get the answer.

**-Write a Graph from a Linear Function Expression**

We have discussed how to get an expression from a graph. On the other hand, how do we draw a graph?

The method is the same for writing a graph from a linear function. Even if the slope is a fraction, we can write a graph from the rate of change. For example, how would the following equation be written in a graph?

- $y=-\displaystyle\frac{2}{3}x+2$

The intercept is 2, so it goes through $(0,2)$. Also, if $x$ increases by 3, $y$ decreases by 2. That is, if $x=3$, then $y=0$. Since the straight line passes through $(3,0)$, we can write the following graph.

A linear function is completed by connecting the two coordinates with a straight line, as mentioned earlier. Even if the slope is a fraction, the way the graph is written is the same.

## Find the Equation for a Linear Function from Intercept and Coordinates

In linear function problems, we may need to find the expression for a linear function from a graph or coordinates. How can we find the equation for a linear function? There are several ways to do this, the easiest method is to find a linear equation from the intercept and coordinates.

In some problems, the value of the intercept is already known. In that case, we can assign the value of $b$.

Also, if we know the coordinates, we know the values of $x$ and $y$. Substituting $x$ determines the value of $y$, which is a linear function. We can find the value of $y$ by assigning any $x$. Therefore, substituting the coordinates will yield a slope.

For example, what is the expression for the linear function in the following case?

- The intercept is 1: $b=1$
- Through the coordinates $(2,2)$

Since the intercept is 1, the equation is $y=ax+1$. And since it passes through the coordinates $(2,2)$, let’s substitute $x=2$ and $y=2$. We get the following.

$2=2a+1$

$-2a=1-2$

$-2a=-1$

$a=\displaystyle\frac{1}{2}$

The slope is $a=\displaystyle\frac{1}{2}$. Therefore, the expression of the linear equation is $y=\displaystyle\frac{1}{2}x+1$.

From the problem sentence, we already know that the intercept is 1. In contrast, we don’t know the slope of the linear function. So we can assign the coordinates to it and get the slope like this.

### Find the Equation for a Linear Function from a Graph

In addition, we may have to come up with an expression for a linear function from the graph. When reading a graph, check the intercept and coordinates of the graph. For example, what is the intercept and coordinates of the following graph?

Checking the $y$ coordinate, the intercept is 1. Also, from the graph, we can assign any coordinates. It doesn’t matter which coordinates you assign. However, in mathematics, we need to use simple coordinates to avoid miscalculation. So, in this case, we will use the coordinates of $(1,-1)$.

From the graph, the intercept is 1, so the expression of the linear function is $y=ax+1$. Also, this linear function passes through the coordinates of $(1,-1)$. So let’s substitute $x=1$ and $y=-1$. The calculation is as follows.

$-1=a+1$

$-a=1+1$

$-a=2$

$a=-2$

The slope is -2, so we get the equation $y=-2x+1$. Even if we need to read the intercept and coordinates from the graph, how to make an equation for a linear function is the same.

Note that it is also possible to calculate $a$ from the rate of change when reading the graph. In the previous graph, as shown below, for every 1 increase in the value of $x$, the value of $y$ is increased by -2. Therefore, we can see that the rate of change is -2.

The rate of change is the same as the slope. From the intercept and the slope, we can see that the equation is $y=-2x+1$.

### Calculating an Equation for a Linear Function in a System of Equations

On the other hand, there are cases in which we are not presented with a graph and do not know the intercept. Only two coordinates are known. For example, if only the following two coordinates are known.

- $(-6,-7)$
- $(3,-1)$

How can we come up with an expression for a linear function? In this case, we can use systems of linear equations (or simultaneous equations) to come up with an equation for a linear function. Even if there are two unknown numbers, we can find the number by making two equations, which is a system of equations.

If we substitute $(-6,-7)$ and $(3,-1)$ into $y=ax+b$, we get the following.

$\begin{eqnarray} \left\{\begin{array}{l}-7=-6a+b\\-1=3a+b\end{array}\right.\end{eqnarray}$

Let’s solve this with simultaneous equations. The result is as follows.

$\begin{array}{r}-7=-6a+b\\\underline{-)\phantom{0}-1=3a+b}\\-6=-9a\\a=\displaystyle\frac{2}{3}\end{array}$

The systems of linear equations give us $a=\displaystyle\frac{2}{3}$. And by solving the simultaneous equations by substituting $a=\displaystyle\frac{2}{3}$, we can calculate $b=-3$.

For example, substituting $(3,-1)$ yields the following.

$-1=3×\displaystyle\frac{2}{3}+b$

$-1=2+b$

$b=-3$

Therefore, $y=\displaystyle\frac{2}{3}x-3$. Even if the graph is not presented, if we know the two coordinates, we can still come up with a linear function by using a system of linear equations.

## How to Find the Intersection of Coordinates Using Linear Functions

Intersection is an application of linear functions. We are often asked to find the intersection of two linear functions. How can we find the intersection point?

For example, try to find the intersection point P of the following two expressions.

- $y=-2x-2$
- $y=x+2$

It is a system of equations that produces a value that satisfies the conditions of two equations. Let’s solve the following system of equations using $y=-2x-2$ and $y=x+2$. Then we can get the intersection.

$\begin{eqnarray} \left\{\begin{array}{l}y=-2x-2\\y=x+2\end{array}\right.\end{eqnarray}$

Why can we find an intersection using a system of linear equations? Using simultaneous equations is the same as finding the values of $x$ and $y$ that satisfy $y=-2x-2$ and $y=x+2$. Calculating $x$ and $y$ that satisfy the two equations is the same as finding the intersection point.

As mentioned above, even if we have two unknown numbers, we can still come up with a number using simultaneous equations. If we solve the system of linear equations then we get the following.

$\begin{array}{r}y=-2x-2\\\underline{-)\phantom{0}\phantom{0}y=x+2}\\0=-3x-4\\3x=-4\\x=-\displaystyle\frac{4}{3}\end{array}$

Then, substitute $x=-\displaystyle\frac{4}{3}$. For example, if we substitute $y=x+2$, we get the following.

$y=-\displaystyle\frac{4}{3}+2$

$y=-\displaystyle\frac{4}{3}+\displaystyle\frac{6}{3}$

$y=\displaystyle\frac{2}{3}$

Thus we know that the coordinates of the point P are $\left(-\displaystyle\frac{4}{3},\displaystyle\frac{2}{3}\right)$.

## Exercises: Linear Function Graphs and Word Problems

**Q1.** Solve the following problems.

- Find the equation for the linear function of (1)
- Find the equation for the linear function of (2)
- Find the intersection of (1) and (2)

**A1.** Answers.

**(a)**

We need to find a linear equation from the graph. So, let’s focus on the intercept. In equation (1), the intercept is 4. Therefore, $b=4$.

Also, (1) passes through the coordinates of $(1,2)$. Therefore, let’s substitute $y=ax+4$. We get the following.

$2=a+4$

$a=-2$

We found that the slope is -2. Therefore, equation (1) is $y=-2x+4$.

**(b)**

The graph does not reveal the intercept in (2). So, let’s calculate equation (2) from two coordinates. Choose any of the (2) coordinates. For example, (2) goes through the following coordinates.

- $(2,3)$
- $(5,4)$

Next, let’s calculate the slope and intercept by a system of equations. Substituting into the formula $y=ax+b$, we get the following.

$\begin{eqnarray} \left\{\begin{array}{l}3=2a+b\\4=5a+b\end{array}\right.\end{eqnarray}$

Solving the simultaneous equations, we get the following.

$\begin{array}{r}3=2a+b\\\underline{-)\phantom{0}4=5a+b}\\-1=–3a\\a=\displaystyle\frac{1}{3}\end{array}$

Substituting $a=\displaystyle\frac{1}{3}$ into $3=2a+b$ gives the following.

$3=2×\displaystyle\frac{1}{3}+b$

$\displaystyle\frac{9}{3}=\displaystyle\frac{2}{3}+b$

$b=\displaystyle\frac{7}{3}$

Therefore, the equation in (2) is $y=\displaystyle\frac{1}{3}x+\displaystyle\frac{7}{3}$.

**(c)**

To get the intersection of the two equations, solve the simultaneous equations. Since we have produced linear functions for (1) and (2), we can use these two equations to create the following system of equations.

$\begin{eqnarray} \left\{\begin{array}{l}y=-2x+4\\y=\displaystyle\frac{1}{3}x+\displaystyle\frac{7}{3}\end{array}\right.\end{eqnarray}$

Using the substitution method, we get the following.

$-2x+4=\displaystyle\frac{1}{3}x+\displaystyle\frac{7}{3}$

$(-2x+4)\textcolor{red}{×3}=\left(\displaystyle\frac{1}{3}x+\displaystyle\frac{7}{3}\right)\textcolor{red}{×3}$

$-6x+12=x+7$

$-6x-x=7-12$

$-7x=-5$

$x=\displaystyle\frac{5}{7}$

Then substitute $x=\displaystyle\frac{5}{7}$ for $y=-2x+4$. The result is as follows.

$y=-2×\displaystyle\frac{5}{7}+4$

$y=-\displaystyle\frac{10}{7}+\displaystyle\frac{28}{7}$

$y=\displaystyle\frac{18}{7}$

Thus the intersection of (1) and (2) can be calculated as $\left(\displaystyle\frac{5}{7},\displaystyle\frac{18}{7}\right)$.

**Q2.** Solve the following problems.

There is a triangle with AB=10cm, BC=6cm, and ∠ABC=90°. Point P departs from A and travels through B at a speed of 1 cm per second to C. Set the area of △APC as $y$ cm^{2}, $x$ seconds after point P departs from A.

- Calculate the area of △APC after 8 seconds.
- For the $x$ and $y$ domains, represent them with inequality signs.
- When the point P is on BC, equate the relationship between $x$ and $y$.

**A2.** Answers.

**(a)**

After 8 seconds, the length of the AP is 8 cm. That is, the horizontal length of the triangle is 8 cm. On the other hand, the length of the vertical is 6 cm. The formula for the area of the triangle is as follows.

- The area of a triangle = Width × Height × $\displaystyle\frac{1}{2}$

Therefore, the area of the triangle is as follows.

- $8×6×\displaystyle\frac{1}{2}=24$

The answer is 24 cm^{2}.

**(b)**

The total length of AB and BC is 16 cm. Since point P travels 1 cm per second, it arrives at C after 16 seconds. Therefore, the domain of $x$ is as follows.

- $0≤x≤16$

On the other hand, for the area of △APC, when point P arrives at B, the area of △APC is at its maximum. This is because when point P travels over BC, the area of △APC is reduced. The area of △ABC is as follows.

- $10×6×\displaystyle\frac{1}{2}=30$

Therefore, the domain of $y$ is as follows.

- $0≤y≤30$

**(c)**

When the point P is on BC, the horizontal length is fixed at 10 cm. On the other hand, the length of the vertical varies. If we can find the length of the vertical (length of CP), we can create an equation that expresses the area of △APC.

How do we figure out the length of the CP? It is difficult to think in terms of a triangle. So, let’s consider the length of the CP in the following straight line.

The length of the AC is 16 cm. And since it moves at 1 cm per second, the length of the AP after $x$ seconds is $x$ cm. That is, the length of the CP is $(16-x)$.

The length of the horizontal is 10 cm and the length of the vertical is $(16-x)$ cm. Therefore, the area of △APC is as follows.

$y=10×(16-x)×\displaystyle\frac{1}{2}$

$y=5×(16-x)$

$y=-5x+80$

## Create and Solve an Equation for Linear Functions

A very similar field to direct proportionality is linear functions. Think of linear functions in mathematics as a special case of a proportional.

There are so many situations where we have to use linear functions to make calculations. For example, when you calculate your future savings, you use linear functions. When calculating, we must find the slope and intercept to create an equation.

In linear functions, there are also many graph-based problems. So you need to be able to calculate the slope and intercept from the graph. By using coordinates, we can get an expression for a linear function. Also, by using simultaneous equations, we can find the intersection of two lines.

Once we understand the properties of these linear functions, we will be able to solve word problems. If you can solve word problems, you will be able to use mathematics in all aspects of your daily life.