Solving Equations 10(q+4)=0 A Step-by-Step Guide
The equation presented is a linear equation in one variable, specifically q. Our goal is to isolate q on one side of the equation to determine its value. This involves using algebraic manipulations to simplify the equation and solve for q. Let's break down the steps involved in solving this equation.
Step-by-Step Solution
The equation we need to solve is:
10(q + 4) = 0
Step 1: Distribute the 10
The first step in solving the equation is to distribute the 10 across the terms inside the parentheses. This means we multiply both q and 4 by 10:
10 * q + 10 * 4 = 0
This simplifies to:
10q + 40 = 0
Step 2: Isolate the term with q
Next, we want to isolate the term containing q (which is 10q) on one side of the equation. To do this, we subtract 40 from both sides of the equation. This maintains the balance of the equation:
10q + 40 - 40 = 0 - 40
This simplifies to:
10q = -40
Step 3: Solve for q
Now, to solve for q, we need to get q by itself. We can do this by dividing both sides of the equation by 10:
10q / 10 = -40 / 10
This simplifies to:
q = -4
Verification
To ensure our solution is correct, we can substitute the value of q back into the original equation and check if it holds true.
Original equation:
10(q + 4) = 0
Substitute q = -4:
10(-4 + 4) = 0
Simplify inside the parentheses:
10(0) = 0
Multiply:
0 = 0
Since the equation holds true, our solution q = -4 is correct.
An alternative approach to solving this equation involves dividing both sides by 10 before distributing. This can sometimes simplify the process.
Step-by-Step Solution (Alternative Method)
Starting with the original equation:
10(q + 4) = 0
Step 1: Divide both sides by 10
Divide both sides of the equation by 10:
(10(q + 4)) / 10 = 0 / 10
This simplifies to:
q + 4 = 0
Step 2: Isolate q
To isolate q, subtract 4 from both sides of the equation:
q + 4 - 4 = 0 - 4
This simplifies to:
q = -4
As we can see, this method also leads us to the same solution, q = -4.
Several key algebraic concepts were applied in solving this equation:
- Distributive Property: This property states that a(b + c) = ab + ac. We used this to expand the expression 10(q + 4).
- Inverse Operations: To isolate the variable, we used inverse operations. For example, we subtracted 40 (the inverse of adding 40) and divided by 10 (the inverse of multiplying by 10).
- Maintaining Equality: It's crucial to perform the same operation on both sides of the equation to maintain equality. This ensures that the equation remains balanced and the solution remains valid.
- Substitution: To verify our solution, we substituted the value of q back into the original equation. If the equation holds true, the solution is correct.
When solving equations like this, it's important to avoid common mistakes:
- Incorrect Distribution: Ensure that you correctly distribute the number outside the parentheses to all terms inside. For example, 10(q + 4) should be 10q + 40, not 10q + 4.
- Not Maintaining Equality: Always perform the same operation on both sides of the equation. If you subtract a number from one side, you must subtract the same number from the other side.
- Arithmetic Errors: Double-check your calculations to avoid simple arithmetic errors, such as incorrect addition or subtraction.
- Forgetting the Sign: Pay close attention to the signs of the numbers, especially when dealing with negative numbers. For example, -40 / 10 is -4, not 4.
Linear equations like this one have numerous real-world applications. They can be used to model various situations, such as:
- Calculating Costs: For example, if you have a fixed cost and a variable cost per item, you can use a linear equation to determine the total cost.
- Determining Break-Even Points: Businesses use linear equations to find the break-even point, where revenue equals expenses.
- Mixing Solutions: In chemistry, linear equations can be used to determine the amounts of different solutions needed to achieve a desired concentration.
- Distance, Rate, and Time Problems: Linear equations are often used to solve problems involving distance, rate, and time, using the formula distance = rate * time.
In summary, we have solved the equation 10(q + 4) = 0 for q. By distributing, isolating the variable, and using inverse operations, we found that q = -4. We also verified our solution by substituting it back into the original equation. Additionally, we explored an alternative method of solving the equation by dividing first, which also yielded the same result. Understanding the key concepts and avoiding common mistakes are essential for solving linear equations effectively. These equations are fundamental in mathematics and have wide-ranging applications in various fields.
The solution to the equation 10(q + 4) = 0 is q = -4.
q = -4
