Solving Linear Equations Step-by-Step Guide

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In the realm of mathematics, linear equations stand as fundamental building blocks. Mastering the art of solving these equations is crucial for success in algebra and beyond. This article serves as a comprehensive guide, walking you through the process of solving various linear equations, step by step. We'll tackle equations involving parentheses, decimals, and variables on both sides, providing clear explanations and strategies to help you confidently conquer any linear equation that comes your way.

Question 9: Solving Equations with Parentheses

Our first equation, 5(x - 8) = 50, presents a classic scenario where we need to deal with parentheses. The key to unlocking this equation lies in the distributive property. This property states that a(b + c) = ab + ac. Applying this to our equation, we multiply the 5 by both terms inside the parentheses.

Step 1: Distribute

Begin by distributing the 5 across the terms inside the parentheses:

5 * x - 5 * 8 = 50

This simplifies to:

5x - 40 = 50

Step 2: Isolate the Variable Term

Our goal is to isolate the term with the variable (5x in this case). To do this, we need to get rid of the -40 on the left side. We achieve this by adding 40 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain the balance of the equation:

5x - 40 + 40 = 50 + 40

This simplifies to:

5x = 90

Step 3: Solve for the Variable

Now we have 5x = 90. To isolate x, we need to undo the multiplication by 5. We do this by dividing both sides of the equation by 5:

5x / 5 = 90 / 5

This gives us:

x = 18

Therefore, the solution to the equation 5(x - 8) = 50 is x = 18. The correct answer is D. 018.

Understanding the distributive property and the importance of performing the same operation on both sides of the equation are vital for solving equations with parentheses. Practice with similar problems to solidify your understanding of this concept.

Question 10: Tackling Equations with Decimals

Next, we encounter an equation with decimals: 0.2(x + 14) = 22. Decimals can sometimes make equations seem intimidating, but the process of solving them is the same as with whole numbers. We can still employ the distributive property and isolate the variable. There are mainly two ways to solve this question. First, we can use distribution and second to remove the decimals by multiplying both sides by 10.

Method 1: Distribute

Step 1: Distribute

Similar to the previous equation, we start by distributing the 0.2 across the terms inside the parentheses:

  1. 2 * x + 0.2 * 14 = 22

This simplifies to:

  1. 2x + 2.8 = 22

Step 2: Isolate the Variable Term

To isolate the term with the variable (0.2x), we subtract 2.8 from both sides of the equation:

  1. 2x + 2.8 - 2.8 = 22 - 2.8

This simplifies to:

  1. 2x = 19.2

Step 3: Solve for the Variable

Now we have 0.2x = 19.2. To isolate x, we divide both sides of the equation by 0.2:

  1. 2x / 0.2 = 19.2 / 0.2

This gives us:

x = 96

Method 2: Multiply by 10

Step 1: Multiply by 10

Multiply both sides of the equation by 10 to eliminate the decimal:

10 * [0.2(x + 14)] = 10 * 22

This simplifies to:

  1. (x + 14) = 220

Step 2: Distribute

Distribute the 2 across the terms inside the parentheses:

    • x + 2 * 14 = 220

This simplifies to:

  1. x + 28 = 220

Step 3: Isolate the Variable Term

To isolate the term with the variable (2x), we subtract 28 from both sides of the equation:

  1. x + 28 - 28 = 220 - 28

This simplifies to:

  1. x = 192

Step 4: Solve for the Variable

Now we have 2x = 192. To isolate x, we divide both sides of the equation by 2:

  1. x / 2 = 192 / 2

This gives us:

x = 96

Therefore, the solution to the equation 0.2(x + 14) = 22 is x = 96. The correct answer is D. 096.

An alternative approach to solving decimal equations is to eliminate the decimal by multiplying both sides of the equation by a power of 10. In this case, multiplying by 10 removes the decimal point. This method can simplify the arithmetic and make the equation easier to solve.

Question 11: Equations with Variables on Both Sides

Our next challenge is the equation 2.5a + 12 = 5a. This equation introduces a new element: variables on both sides of the equation. The key to solving such equations is to gather the variable terms on one side and the constant terms on the other. When we have variables on both sides of the equation, we can move the variables to one side and the constants to the other side.

Step 1: Gather Variable Terms

To begin, we want to get all the 'a' terms on one side. We can subtract 2.5a from both sides of the equation:

  1. 5a + 12 - 2.5a = 5a - 2.5a

This simplifies to:

12 = 2.5a

Step 2: Solve for the Variable

Now we have 12 = 2.5a. To isolate 'a', we divide both sides of the equation by 2.5:

  1. 5 / 2.5 = 2.5a / 2.5

This gives us:

a = 4.8

Therefore, the solution to the equation 2.5a + 12 = 5a is a = 4.8. The correct answer is C. 04.8.

Remember, the goal is to isolate the variable. By strategically adding or subtracting terms from both sides, we can rearrange the equation to achieve this goal.

Question 12: More Practice with Variables on Both Sides

Let's tackle another equation with variables on both sides: 14b - 22 = 11b. This equation reinforces the techniques we learned in the previous example. Isolating the variables is the key here.

Step 1: Gather Variable Terms

To get all the 'b' terms on one side, we can subtract 11b from both sides of the equation:

14b - 22 - 11b = 11b - 11b

This simplifies to:

  1. b - 22 = 0

Step 2: Isolate the Variable Term

Next, we need to isolate the term with the variable (3b). To do this, we add 22 to both sides of the equation:

  1. b - 22 + 22 = 0 + 22

This simplifies to:

  1. b = 22

Step 3: Solve for the Variable

Now we have 3b = 22. To isolate 'b', we divide both sides of the equation by 3:

  1. b / 3 = 22 / 3

This gives us:

b = 7.33

Therefore, the solution to the equation 14b - 22 = 11b is b = 7.33. The answer was not provided in the options.

By consistently applying these steps, you can confidently solve linear equations with variables on both sides. Remember to focus on isolating the variable by performing the same operations on both sides of the equation.

Conclusion: Mastering Linear Equations

Solving linear equations is a fundamental skill in algebra. By understanding the principles of the distributive property, isolating variables, and dealing with decimals, you can confidently tackle a wide range of equations. Remember to practice regularly, and don't be afraid to break down complex equations into smaller, manageable steps. With dedication and a solid understanding of these concepts, you'll be well-equipped to excel in your mathematical journey.