One-Step Equations

One-step equations are algebraic expressions that are used to solve for the unknown variable in just a single step. To solve these equations, we need to isolate the variable to one side and the constants to the other side of the equal sign.

The mathematical operations required to solve a one-step equation are:

Example: x + 2 = 10
x = 8, where x is the unknown variable.

In the above equation, when 2 was moved to the other side, its sign changed from positive to negative.
So, 10 - 2 = 8.

A one-step equation is all about applying the inverse of the operation given in the equation.
For example, if the given operation is addition, then we should change the operation to subtraction while solving a one-step equation.

Here are some common operations and their inverse function: 

Now, let's understand the step-by-step breakdown of solving one-step equations:

1. Determine the operation implemented on the variable.
2. Identify its inverse operation using the given chart.
3. Perform the inverse operation on both side of the equation.

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To solve a one-step equation that has either addition or subtraction as its operation, we need to follow a basic rule.

Rule: Whenever a constant or variable is shifted from either the RHS to the LHS or vice versa, its operation changes to its inverse.

For example
 

1. In the equation x - 2 = 8, when -2 is shifted from LHS to RHS, the equation becomes x = 8 + 2. Now, we have isolated x and can proceed to find its value, which is 10.  
    
2. Equation is x + 2 = 8 can be written as x = 8 - 2.
   Here, we have isolated x, and while shifting + 2 from the LHS of the equation to the RHS, its operation changes from +2 to -2.
   So the value of x is: x = 8 - 2 = 6. 

Let's practice this using a word problem.

Practice Problem: If you bought a pen for $5 then, and you have $6 left, now how much money did you have initially?

Solution: To solve this, we can form an equation, assuming x to be the total money.

x - $5 = $6

• Adding 5 on both side.
  \(x - $5 = $6 + $5 \\ x = $11\)

You had $11 initially.

To solve a one-step equation involving multiplication or division, just change the operation to its inverse while isolating the variable.

• If the variable is multiplied, we will divide the whole equation with the same number.

• If the variable is divided, we will multiply both sides by the same number.

Let's understand this using some examples.

Example:

1. Solve: 5x = 25
   
   Here, 5 is multiplied, so we divide both sides by 5.
   \(5x ÷ 5 = 25 ÷ 5\\ x = 5\)
    
2. Solve: x ÷ 7 = 2
   
   Since x is divided by 7, multiply both sides by 7.
   \(x ÷ 7 × 7 = 2 × 7\\ x = 14\)

Let's practice this using a word problem

Practice Problem: If you bought 5 chocolates and the total price is $20, then what is the price of 1 chocolate.

Solution: Suppose the price of 1 chocolate is x.

5x = 20

• Since, 5 is multiplied by the variable, we will divide the equation by 5
  \({5x \over 5} = {20 \over 5} \\ x = $4\)

The price of one chocolate is $4.

To help you master the concept of one-step equations, let's learn some few essential tips and tricks:

1. Remember, balance is the key. Whatever changes you do on one side, also do it on the another.
    
2. Think of inverse operation as undoing. If a number is added, then undo it, implies subtract it from the variable.
    
3. Memorize the inverse operations correctly.
    
4. To check if your answer is correct, substitute it in the equation.
    
5. Carefully perform the multiplication and division. You can use multiplication table and division table also.
    

Parent Tip: Encourage your child to practice problems. Use real life objects or items to related for better visualization.

One-step equations are not only used in solving math problems; they are also useful in everyday life. These equations help us find the unknown values quickly by performing the basic arithmetic operations. Here are some of the real-life examples where one-step equations are used.

1. Shopping and Budgeting: While shopping and budgeting, if we know how much we spent and how much money we had at first, we can find out how much money is left.
   
   For example, if we had $500 before purchasing a bag for $120, we would have $380 remaining. How did we land at 380? Let’s look at the equation below. 
   
   Equation: x = $500 -  $120
   x = $380
   
    
2. Cooking and Recipes: We can use equations to adjust the amount of ingredients while cooking.
   
   For example, If you need 3 cups of flour for 1 cake, then how much flour do you need for 4 cakes?
   
   Equation: x = 3 × 4
   x = 12 cups
   
    
3. Distance or Speed: One-step equations can help us find speed when distance and time are known.
   
   For example, if you travel 60 km in 3 hours, what is your speed per hour? To solve this problem, we use the formula: Speed = Distance/Time
   
   So the equation is x = 60 ÷ 3
   x = 20 km/h.
   
    
4. Counting Items: Equations can be used to count the number of objects or items when quantities are added or reduced.
   
   For example, If you had 5 cookies, and you ate 3 of them, the number of cookies left can be given by the following equation:
   
   Equation: x = 5 - 3
   x = 2
   
    
5. Area or Volume: We can find the area, perimeter or volume of any shape using equations if the measurements are known.
   
   For example, the area of a square can be given as side². If the side is 3 cm, then the area can be calculated as:
   
   Equation, x = 3²
   x = 9