Solving For X: A Step-by-Step Guide To Linear Equations

Welcome to the world of solving linear equations! This guide will walk you through ten different equations, providing a detailed, step-by-step solution for each. Whether you're a student brushing up on algebra or just looking to sharpen your math skills, this article will help you master the art of solving for x. So, let's dive in and conquer these equations together!

1. Solving the equation $3x + 8 = 7x - 8$

In this section, we'll tackle the equation $3x + 8 = 7x - 8$. Our primary goal is to isolate x on one side of the equation. Here’s how we do it:

1. Combine like terms: First, let's gather all the 'x' terms on one side and the constants on the other. Subtract $3x$ from both sides of the equation:

   $3x + 8 - 3x = 7x - 8 - 3x$

   This simplifies to:

   $8 = 4x - 8$

2. Isolate the x term: Now, we want to isolate the term with 'x'. Add 8 to both sides:

   $8 + 8 = 4x - 8 + 8$

   This gives us:

   $16 = 4x$

3. Solve for x: Finally, to solve for 'x', divide both sides by 4:

   $16 / 4 = 4x / 4$

   Therefore:

   $x = 4$

So, the solution to the equation $3x + 8 = 7x - 8$ is $x = 4$. This step-by-step approach ensures clarity and accuracy in finding the value of x.

2. Solving the equation $7x + 5 = 5x + 1$

Let's solve the equation $7x + 5 = 5x + 1$. The key here is to manipulate the equation to get 'x' alone on one side. Here’s the breakdown:

1. Combine like terms: Start by subtracting $5x$ from both sides to get the 'x' terms together:

   $7x + 5 - 5x = 5x + 1 - 5x$

   This simplifies to:

   $2x + 5 = 1$

2. Isolate the x term: Next, subtract 5 from both sides to isolate the term with 'x':

   $2x + 5 - 5 = 1 - 5$

   This gives us:

   $2x = -4$

3. Solve for x: Finally, divide both sides by 2 to solve for 'x':

   $2x / 2 = -4 / 2$

   Therefore:

   $x = -2$

The solution to the equation $7x + 5 = 5x + 1$ is $x = -2$. By systematically isolating 'x', we arrive at the solution in a clear and understandable manner.

3. Solving the equation $5x + 7 = 9x + 1$

Now, let's solve $5x + 7 = 9x + 1$. Our goal remains the same: isolate 'x' to find its value. Here’s how:

1. Combine like terms: Subtract $5x$ from both sides to gather the 'x' terms:

   $5x + 7 - 5x = 9x + 1 - 5x$

   This simplifies to:

   $7 = 4x + 1$

2. Isolate the x term: Subtract 1 from both sides to isolate the term with 'x':

   $7 - 1 = 4x + 1 - 1$

   This gives us:

   $6 = 4x$

3. Solve for x: Divide both sides by 4 to solve for 'x':

   $6 / 4 = 4x / 4$

   Which simplifies to:

   $x = 3/2$ or $x = 1.5$

Thus, the solution to the equation $5x + 7 = 9x + 1$ is $x = 1.5$. Each step is designed to bring us closer to the value of 'x'.

4. Solving the equation $4x + 3 = 7 - x$

Let's tackle the equation $4x + 3 = 7 - x$. The approach involves rearranging the equation to isolate 'x'. Here's the step-by-step solution:

1. Combine like terms: Add 'x' to both sides to get all 'x' terms on one side:

   $4x + 3 + x = 7 - x + x$

   This simplifies to:

   $5x + 3 = 7$

2. Isolate the x term: Subtract 3 from both sides to isolate the term with 'x':

   $5x + 3 - 3 = 7 - 3$

   This gives us:

   $5x = 4$

3. Solve for x: Divide both sides by 5 to solve for 'x':

   $5x / 5 = 4 / 5$

   Therefore:

   $x = 4/5$ or $x = 0.8$

The solution to the equation $4x + 3 = 7 - x$ is $x = 0.8$. This systematic isolation ensures accurate calculation of 'x'.

5. Solving the equation $15x - 4 = 10 - 3x$

Now, let's solve $15x - 4 = 10 - 3x$. The process involves combining 'x' terms and constants to find the value of 'x'. Follow these steps:

1. Combine like terms: Add $3x$ to both sides to get all 'x' terms on one side:

   $15x - 4 + 3x = 10 - 3x + 3x$

   This simplifies to:

   $18x - 4 = 10$

2. Isolate the x term: Add 4 to both sides to isolate the term with 'x':

   $18x - 4 + 4 = 10 + 4$

   This gives us:

   $18x = 14$

3. Solve for x: Divide both sides by 18 to solve for 'x':

   $18x / 18 = 14 / 18$

   Which simplifies to:

   $x = 7/9$ or approximately $x = 0.78$

Therefore, the solution to the equation $15x - 4 = 10 - 3x$ is approximately $x = 0.78$. Each algebraic manipulation brings us closer to the accurate value of 'x'.

6. Solving the equation $5(x + 1) = 4(x + 2)$

Let's solve $5(x + 1) = 4(x + 2)$. This equation requires us to first distribute the numbers outside the parentheses. Here’s how:

1. Distribute: Expand both sides of the equation:

   $5 * x + 5 * 1 = 4 * x + 4 * 2$

   This simplifies to:

   $5x + 5 = 4x + 8$

2. Combine like terms: Subtract $4x$ from both sides to gather the 'x' terms:

   $5x + 5 - 4x = 4x + 8 - 4x$

   This simplifies to:

   $x + 5 = 8$

3. Isolate the x term: Subtract 5 from both sides to isolate 'x':

   $x + 5 - 5 = 8 - 5$

   Therefore:

   $x = 3$

The solution to the equation $5(x + 1) = 4(x + 2)$ is $x = 3$. Distributing correctly is crucial for accurately solving equations of this type.

7. Solving the equation $8(x + 5) = 10(x + 3)$

Now, let’s solve $8(x + 5) = 10(x + 3)$. Similar to the previous equation, we start by distributing the numbers outside the parentheses. Here’s the step-by-step process:

1. Distribute: Expand both sides of the equation:

   $8 * x + 8 * 5 = 10 * x + 10 * 3$

   This simplifies to:

   $8x + 40 = 10x + 30$

2. Combine like terms: Subtract $8x$ from both sides to get the 'x' terms together:

   $8x + 40 - 8x = 10x + 30 - 8x$

   This simplifies to:

   $40 = 2x + 30$

3. Isolate the x term: Subtract 30 from both sides to isolate the term with 'x':

   $40 - 30 = 2x + 30 - 30$

   This gives us:

   $10 = 2x$

4. Solve for x: Divide both sides by 2 to solve for 'x':

   $10 / 2 = 2x / 2$

   Therefore:

   $x = 5$

The solution to the equation $8(x + 5) = 10(x + 3)$ is $x = 5$. Careful distribution and simplification lead to the correct value of 'x'.

8. Solving the equation $3(x - 5) = 7(x + 4) - 7$

Let’s tackle $3(x - 5) = 7(x + 4) - 7$. This equation combines distribution with simplification of constants. Here’s the breakdown:

1. Distribute: Expand both sides of the equation:

   $3 * x - 3 * 5 = 7 * x + 7 * 4 - 7$

   This simplifies to:

   $3x - 15 = 7x + 28 - 7$

2. Simplify: Combine the constants on the right side:

   $3x - 15 = 7x + 21$

3. Combine like terms: Subtract $3x$ from both sides to gather the 'x' terms:

   $3x - 15 - 3x = 7x + 21 - 3x$

   This simplifies to:

   $-15 = 4x + 21$

4. Isolate the x term: Subtract 21 from both sides to isolate the term with 'x':

   $-15 - 21 = 4x + 21 - 21$

   This gives us:

   $-36 = 4x$

5. Solve for x: Divide both sides by 4 to solve for 'x':

   $-36 / 4 = 4x / 4$

   Therefore:

   $x = -9$

The solution to the equation $3(x - 5) = 7(x + 4) - 7$ is $x = -9$. Accurate distribution and constant simplification are key to solving this equation correctly.

9. Solving the equation $3.1(4.8x - 1) - 3.9 = x + 1$

Now, let’s solve $3.1(4.8x - 1) - 3.9 = x + 1$. This equation involves decimals, so precision is crucial. Here’s the step-by-step solution:

1. Distribute: Expand the left side of the equation:

   $3. 1 * 4.8x - 3.1 * 1 - 3.9 = x + 1$

   This simplifies to:

   $14.88x - 3.1 - 3.9 = x + 1$

2. Simplify: Combine the constants on the left side:

   $14.88x - 7 = x + 1$

3. Combine like terms: Subtract 'x' from both sides to gather the 'x' terms:

   $14.88x - 7 - x = x + 1 - x$

   This simplifies to:

   $13.88x - 7 = 1$

4. Isolate the x term: Add 7 to both sides to isolate the term with 'x':

   $13.88x - 7 + 7 = 1 + 7$

   This gives us:

   $13.88x = 8$

5. Solve for x: Divide both sides by 13.88 to solve for 'x':

   $13.88x / 13.88 = 8 / 13.88$

   Therefore:

   $x ≈ 0.58$

The solution to the equation $3.1(4.8x - 1) - 3.9 = x + 1$ is approximately $x = 0.58$. Decimal calculations require careful attention to detail for accurate results.

10. Solving the equation $8.9(x - 3.5) + 4.2(3x + 2.3) = 4.7x$

Finally, let’s solve $8.9(x - 3.5) + 4.2(3x + 2.3) = 4.7x$. This equation combines distribution, decimals, and multiple terms, making it a comprehensive exercise. Here’s the step-by-step solution:

1. Distribute: Expand both sides of the equation:

   $8. 9 * x - 8.9 * 3.5 + 4.2 * 3x + 4.2 * 2.3 = 4.7x$

   This simplifies to:

   $8.9x - 31.15 + 12.6x + 9.66 = 4.7x$

2. Simplify: Combine like terms on the left side:

   $21. 5x - 21.49 = 4.7x$

3. Combine like terms: Subtract $4.7x$ from both sides to gather the 'x' terms:

   $21. 5x - 21.49 - 4.7x = 4.7x - 4.7x$

   This simplifies to:

   $16.8x - 21.49 = 0$

4. Isolate the x term: Add 21.49 to both sides to isolate the term with 'x':

   $16. 8x - 21.49 + 21.49 = 0 + 21.49$

   This gives us:

   $16.8x = 21.49$

5. Solve for x: Divide both sides by 16.8 to solve for 'x':

   $16. 8x / 16.8 = 21.49 / 16.8$

   Therefore:

   $x ≈ 1.28$

The solution to the equation $8.9(x - 3.5) + 4.2(3x + 2.3) = 4.7x$ is approximately $x = 1.28$. Solving complex equations like this requires careful distribution, simplification, and attention to detail.

Conclusion

Congratulations! You've worked through ten different linear equations and successfully solved for x in each one. By understanding the fundamental principles of combining like terms, isolating variables, and applying the distributive property, you can confidently tackle a wide range of algebraic problems. Keep practicing, and you'll become even more proficient at solving for x! For further learning, check out Khan Academy's Algebra section for more resources.