Solving -2x + 14 + 10x = 34 How To Find X

In the realm of mathematics, solving equations is a fundamental skill. Equations represent a balance, a relationship between variables and constants. Our focus today is on linear equations, where the variable, in this case, x, is raised to the power of 1. We'll dissect the equation -2x + 14 + 10x = 34, guiding you through the steps to isolate x and discover its true value.

Understanding the Equation

Before diving into the solution, let's grasp the essence of the equation -2x + 14 + 10x = 34. It presents a scenario where several terms involving x and constants interact. Our mission is to simplify this equation, bringing all x terms to one side and constants to the other, eventually revealing the value of x that satisfies the equation.

Step 1: Combining Like Terms

The first step in simplifying the equation is to combine like terms. Like terms are those that share the same variable raised to the same power. In our equation, we have two terms involving x: -2x and 10x. By combining these, we get:

-2x + 10x = 8x

Now, our equation transforms into:

8x + 14 = 34

Step 2: Isolating the Variable Term

Our next goal is to isolate the term containing x, which is 8x. To do this, we need to eliminate the constant term, 14, from the left side of the equation. We can achieve this by subtracting 14 from both sides of the equation. Remember, maintaining the balance of the equation is crucial, so whatever operation we perform on one side, we must perform on the other:

8x + 14 - 14 = 34 - 14

This simplifies to:

8x = 20

Step 3: Solving for x

Now, we're in the final stretch. We have 8x = 20, and we want to find the value of a single x. To do this, we'll divide both sides of the equation by the coefficient of x, which is 8:

8x / 8 = 20 / 8

This gives us:

x = 20 / 8

Step 4: Simplifying the Fraction

The solution we have, x = 20 / 8, is a fraction. To express it in its simplest form, we need to find the greatest common divisor (GCD) of the numerator (20) and the denominator (8). The GCD of 20 and 8 is 4. Dividing both the numerator and denominator by 4, we get:

x = (20 / 4) / (8 / 4)

x = 5 / 2

Therefore, the solution for x in the equation -2x + 14 + 10x = 34 is x = 5/2. This corresponds to option C in the given choices.

Verifying the Solution

It's always a good practice to verify our solution. To do this, we substitute x = 5/2 back into the original equation:

-2(5/2) + 14 + 10(5/2) = 34

Simplifying, we get:

-5 + 14 + 25 = 34

34 = 34

Since the equation holds true, our solution x = 5/2 is indeed correct.

While we've solved the equation -2x + 14 + 10x = 34 using a step-by-step approach, let's explore alternative methods that can be employed to tackle similar linear equations. Understanding these methods provides a broader perspective and enhances your problem-solving toolkit.

1. Using the Distributive Property (If Applicable)

In some linear equations, you might encounter parentheses. The distributive property comes into play here. It states that a(b + c) = ab + ac. This property allows you to eliminate parentheses and simplify the equation.

For example, consider the equation:

2(x + 3) - 5 = 11

Applying the distributive property, we get:

2x + 6 - 5 = 11

Now, we can proceed with combining like terms and isolating x as demonstrated earlier.

2. Clearing Fractions or Decimals

Equations involving fractions or decimals can sometimes appear daunting. However, we can simplify them by clearing the fractions or decimals. To clear fractions, we multiply both sides of the equation by the least common multiple (LCM) of the denominators. To clear decimals, we multiply both sides by a power of 10 that will eliminate the decimal points.

Let's illustrate with an example:

(1/2)x + (1/3) = (5/6)

The LCM of 2, 3, and 6 is 6. Multiplying both sides by 6, we get:

6 * [(1/2)x + (1/3)] = 6 * (5/6)

3x + 2 = 5

Now, the equation is free of fractions and easier to solve.

3. Transposition Method

The transposition method offers a shortcut for moving terms across the equals sign. It involves changing the sign of a term when it's moved from one side to the other.

For instance, in the equation:

3x - 7 = 8

To isolate the x term, we can transpose the -7 to the right side, changing its sign:

3x = 8 + 7

3x = 15

Now, we can simply divide both sides by 3 to solve for x.

4. Graphical Method

Linear equations can also be solved graphically. This involves plotting the equation on a coordinate plane and finding the point where the line intersects the x-axis (for equations in the form y = mx + c) or the point of intersection for two linear equations.

While the graphical method might not provide the most precise solution, it offers a visual representation of the equation and its solution.

Linear equations aren't just abstract mathematical concepts; they're powerful tools that help us model and solve real-world problems. From calculating distances and speeds to determining costs and profits, linear equations are woven into the fabric of our daily lives. Let's explore some fascinating applications:

1. Calculating Distance, Speed, and Time

The fundamental relationship between distance, speed, and time is expressed through a linear equation:

Distance = Speed × Time

This equation allows us to solve various scenarios. For example:

• If you travel at a constant speed of 60 miles per hour for 2 hours, the distance you cover can be calculated as:

  Distance = 60 mph × 2 hours = 120 miles

• If you need to travel 300 miles and can maintain an average speed of 50 miles per hour, the time it will take can be determined as:

  Time = Distance / Speed = 300 miles / 50 mph = 6 hours

2. Determining Costs and Profits

Businesses often use linear equations to model costs, revenue, and profits. For instance:

• A company manufactures a product with a fixed cost of $1000 and a variable cost of $5 per unit. The total cost of producing x units can be represented by the linear equation:

  Total Cost = 1000 + 5x

• If the company sells each unit for $12, the revenue generated from selling x units can be represented by the linear equation:

  Revenue = 12x

• The profit can then be calculated as the difference between revenue and total cost:

  Profit = Revenue - Total Cost = 12x - (1000 + 5x) = 7x - 1000

By analyzing these equations, businesses can make informed decisions about pricing, production levels, and profitability.

3. Mixing Solutions

Linear equations are also used in chemistry and other fields to solve mixture problems. For example:

• Suppose you have two solutions of acid, one with a 20% concentration and another with a 40% concentration. You want to mix them to obtain 100 liters of a 25% solution. The amount of each solution needed can be determined using a system of linear equations.

4. Converting Units

Conversion formulas between different units of measurement are often linear equations. For example:

• The conversion between Celsius (°C) and Fahrenheit (°F) is given by the linear equation:

  °F = (9/5)°C + 32

This equation allows us to easily convert temperatures from one scale to another.

Solving linear equations, while a fundamental skill, can be prone to errors if not approached with care. Recognizing common pitfalls is crucial for accurate problem-solving. Let's delve into some frequent mistakes and how to steer clear of them:

1. Incorrectly Combining Like Terms

As we discussed earlier, combining like terms is a cornerstone of simplifying equations. However, errors can creep in if the terms aren't truly "like" or if the operations are performed incorrectly. For instance, consider the expression:

3x + 2y - x + 5

A common mistake is to combine the 3x and 2y terms, which is incorrect because they involve different variables. The correct simplification would be:

(3x - x) + 2y + 5 = 2x + 2y + 5

2. Neglecting the Distributive Property

When dealing with equations involving parentheses, the distributive property is essential. Forgetting to apply it correctly can lead to significant errors. Let's revisit our earlier example:

2(x + 3) - 5 = 11

Failing to distribute the 2 across both terms inside the parentheses would result in an incorrect equation.

3. Forgetting to Perform Operations on Both Sides

The golden rule of equation solving is maintaining balance. Any operation performed on one side of the equation must be mirrored on the other side. Neglecting this principle can lead to a skewed solution. For example, in the equation:

5x + 4 = 19

If we subtract 4 only from the left side, we'll disrupt the balance and arrive at an incorrect value for x.

4. Incorrectly Transposing Terms

The transposition method, while a handy shortcut, requires careful attention to sign changes. Remember, when a term is moved across the equals sign, its sign must be flipped. For example, in the equation:

4x - 9 = 7

Transposing the -9 to the right side requires changing it to +9:

4x = 7 + 9

5. Not Simplifying Fractions or Decimals

As we discussed earlier, fractions and decimals can be simplified to make equations easier to solve. Failing to do so can lead to cumbersome calculations and a higher chance of error. Always aim to express your answer in its simplest form.

6. Skipping Steps

While it's tempting to rush through the solution process, skipping steps can increase the likelihood of making mistakes. Each step serves a purpose, and carefully executing them ensures accuracy.

7. Not Verifying the Solution

Verification is the final safeguard against errors. Substituting your solution back into the original equation confirms its validity. This simple step can catch any mistakes made during the solving process.

In conclusion, solving the linear equation -2x + 14 + 10x = 34 involves combining like terms, isolating the variable, and simplifying the result, ultimately leading to the solution x = 5/2. We've explored alternative methods like using the distributive property, clearing fractions, and transposition. Real-world applications in calculating distances, costs, and mixing solutions highlight the practicality of linear equations. By avoiding common mistakes and diligently verifying solutions, you can confidently navigate the world of linear equations. Remember, practice is key to mastering any mathematical concept. So, keep solving, keep exploring, and keep unlocking the power of equations!