How Can We Spot Linear Equations in Everyday Life?

Linear equations are all around us! Understanding them can help us see how different situations can be expressed using math. So, what is a linear equation?

In simple words, a linear equation looks like this: $y = mx + b$. Here’s what those letters mean:

• $y$ is what we want to find out.
• $m$ is the slope, which tells us how fast $y$ changes.
• $x$ is the input, or what we are using to calculate $y$.
• $b$ is the y-intercept, which is the value of $y$ when $x$ is zero.

Now that we know what a linear equation is, how do we find them in real life? Let’s look at some examples!

1. Budgeting and Spending

Imagine you are saving up for a new video game console that costs £300. If you save £20 each week, you can write a linear equation for your savings. In this case, your savings ($y$) depend on how many weeks ($x$) you save.

Your equation would be:

$y = 20x$

• If you save for 5 weeks, then $y = 20(5) = 100$.
• After 10 weeks, you would have $y = 20(10) = 200$.

As you can see, your savings go up by the same amount each week, which shows a linear equation!

2. Travel and Distance

Think about when you ride your bike at a steady speed. For example, if you are going 12 km/h, the distance ($d$) you ride can be shown with:

$d = 12t$

Here, $t$ is the time in hours.

• After 1 hour, you would travel $d = 12(1) = 12$ km.
• After 3 hours, the distance would be $d = 12(3) = 36$ km.

Again, we see that the distance you travel increases steadily, which is a sign of a linear relationship.

3. Temperature Conversions

You might also notice linear equations when converting temperatures. For instance, to change Celsius to Fahrenheit, you could use this equation:

$F = \frac{9}{5}C + 32$

In this equation, $F$ (Fahrenheit) changes evenly as $C$ (Celsius) changes.

• If it’s 0°C, then $F = \frac{9}{5}(0) + 32 = 32$°F.
• If it’s 20°C, then $F = \frac{9}{5}(20) + 32 = 68$°F.

The temperature goes up in a steady way, showing another linear relationship.

4. Fuel Consumption

Let's talk about cars! If a car uses fuel at a steady rate, like getting 40 miles per gallon, then we can write its distance traveled as:

$d = 40g$

Here, $g$ is the gallons of fuel used.

• If you have 2 gallons, the car can go $d = 40(2) = 80$ miles.
• With 5 gallons, it would travel $d = 40(5) = 200$ miles.

You can see that the distance increases evenly with how much gas you use.

Conclusion

Finding linear equations in everyday life can help you get better at math and solve problems. Whether you’re saving money, traveling, converting temperatures, or looking at fuel use, linear equations give us helpful information about how different things relate to each other. Next time you see something that changes steadily, you might just be looking at a linear equation!