3 Practical Examples of Solving Linear Equations

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.