Two-Step Equations

A type of algebraic problems, known as two-step equations, can be resolved in just two steps. To solve these equations, we perform the same mathematical operation on both sides of the equation to maintain equality. To find the value of the variable on one side of the equation, we isolate it.

Two-step equations are written in the form ax + b = c, where a, b, and c are all real numbers. Here are some examples of two-step equations:

• 2x + 3 = 7 
   
• 0.3y + 5 = 1 
   
• (2/3)z - 12 = 10

Two-step equations usually consist of a series of operations like addition, subtraction, division, or multiplication. To solve 2x + 3 = 11, subtract 3 from each side and divide the result by 2.

This procedure is strengthened by practicing two-step equations using a variety of problems and worksheets. Numerous practice issues and detailed solutions are provided by resources such as this article to help you solve these equations.

To isolate the variables in a two-step equation involving decimals and fractions, we can use two inverse operations, such as addition and subtraction, or multiplication and division. Correct decimal point alignment is crucial when working with decimals, and using common denominators makes fractions easier to understand.

For instance, to find 0.5𝑥 = 2.5 in an equation like 0.5𝑥 + 1.2 = 3.7, you must first subtract 1.2 from both sides and then divide by 0.5 to get 𝑥 = 5. Similarly, adding 3.2 to both sides of an equation like 2.4𝑥 − 3.2 = 4.4 yields 2.4𝑥 = 7.6, and dividing by 2.4 yields an approximate solution of 𝑥 ≈ 3.16.

Apart from securing good grades in exams, two-step equations also help solve real-life situations. In this section, let’s see some of the real-life applications of two-step equations. 

Expenses and Budgeting

You may be aware of the total amount spent and fixed expenses (such as rent) when managing a monthly budget, but you must determine your daily spending. A two-step formula such as 30𝑥 + 500 = 800 can be used to determine your daily spending (x) if your rent is $500 and your total expenses are $800.

Taxes and Savings on Shopping

Let's say you know the total cost of the items you are purchasing after the tax and discount have been applied. For instance, the equation 1.08 (p − 10) = x can be used to determine the original price before tax and discount if a jacket costs x dollars and there is a $10 discount plus 8% sales tax. Here, p is the original price of the jacket and p-10 

Distance and Travel Calculations

A two-step equation can be used to calculate speed or time if you travel at a steady pace and take breaks. For example, the formula 4𝑥 = 240 can be used to determine your speed (x) if you drive 240 miles in 5 hours, including a 1-hour break. This is assuming that you drive for 4 hours. Here, the equation 4x = 240 is derived from the formula Distance = Speed × Time, where the total driving time is 4 hours, distance is 240 miles, and x is the speed. 

Cooking and Modifications to the Recipe

Two-step equations are frequently needed to modify ingredients in recipes that call for doubling or halving. The formula 𝑥 + 0.5 = 2.5 aids in calculating the base amount required; for instance, if a recipe calls for "x" cups of flour plus an extra 0.5 cup for kneading, for a total of 2.5 cups.

Planning and Saving Objectives

You can determine how much more you need to save each month if you have some money saved already and are saving for a goal. For instance, saving x every month for six months results in the equation 6𝑥 + 200 = 500, which you must solve for x if your goal is $500, and you currently have $200.