3 Practical Examples of Solving Linear Equations

Learn how to solve linear equations with these practical examples.
By Taylor

Understanding Linear Equations

Linear equations are mathematical statements that show the equality of two expressions. They typically take the form of ax + b = c, where x is the variable we want to solve for. Solving linear equations is essential in various real-world applications, from budgeting to physics. In this article, we’ll explore three practical examples of solving linear equations.

Example 1: Budgeting for a Party

When planning a party, you want to keep track of your expenses. Let’s say you have a total budget of \(200, and you’ve already spent \)50 on decorations. You want to buy some snacks and drinks, costing $5 each. How many snacks and drinks can you buy?

First, you set up the equation based on your budget:

  • Total Budget: $200
  • Amount Spent: $50
  • Cost per Snack/Drink: $5

The equation to represent this situation is:

$$
50 + 5x = 200
$$

Where x is the number of snacks and drinks. Now, let’s solve for x:

  1. Subtract 50 from both sides:
    $$
    5x = 200 - 50
    $$
    $$
    5x = 150
    $$
  2. Divide both sides by 5:
    $$
    x = \frac{150}{5}
    $$
    $$
    x = 30
    $$

You can buy 30 snacks and drinks for your party.

Notes: If the cost of snacks and drinks changes, adjust the equation accordingly. For example, if each snack/drink costs \(6 instead of \)5, the equation would be 50 + 6x = 200.

Example 2: Calculating Distance Traveled

Imagine you’re planning a road trip and need to know how long it will take you to reach your destination. You plan to drive at a speed of 60 miles per hour and want to cover a distance of 240 miles. You can use a linear equation to determine how many hours your trip will take.

The equation that represents distance is:

$$
Distance = Speed \times Time
$$

Let’s define the variables:

  • Distance: 240 miles
  • Speed: 60 miles/hour
  • Time: t hours

Thus, the equation becomes:

$$
60t = 240
$$

Now, let’s solve for t:

  1. Divide both sides by 60:
    $$
    t = \frac{240}{60}
    $$
    $$
    t = 4
    $$

It will take you 4 hours to reach your destination.

Notes: If you need to include breaks, you can modify the equation by adding a variable for breaks. For example, if you plan to take a 30-minute break, you can adjust the total time to t + 0.5.

Example 3: Finding the Price of an Item

Let’s say you’re shopping for a new jacket. The jacket is on sale for \(80, and you have a coupon that gives you a discount of \)20 off the regular price. You want to find out the regular price of the jacket before the discount.

You can set up the equation as follows:

  • Regular Price: p
  • Discount: $20
  • Sale Price: $80

Thus, the equation is:

$$
p - 20 = 80
$$

Now, let’s solve for p:

  1. Add 20 to both sides:
    $$
    p = 80 + 20
    $$
    $$
    p = 100
    $$

The regular price of the jacket is $100.

Notes: If there are additional discounts or taxes, you can modify the equation to include those factors as well. For example, if there’s a tax of $10, the equation might look different, such as p - 20 + 10 = 80.