10. Solve For \[$ N \$\]:$\[ 24 = 13n - 4n + 9 \\]12. Solve For \[$ C \$\]:$\[ \frac{7}{10}c - 8 - \frac{1}{2}c = -16 \\]
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving two linear equations, one involving a single variable and the other involving a fraction. We will break down the steps to solve these equations and provide examples to illustrate the process.
Solving the First Equation: 24 = 13n - 4n + 9
The first equation is 24 = 13n - 4n + 9. To solve for n, we need to isolate the variable n on one side of the equation.
Step 1: Combine like terms
The first step is to combine the like terms on the right-hand side of the equation. In this case, we have two terms involving n, which are 13n and -4n. We can combine these terms by adding their coefficients.
13n - 4n = 9n
So, the equation becomes:
24 = 9n + 9
Step 2: Subtract 9 from both sides
Next, we need to get rid of the constant term on the right-hand side of the equation. We can do this by subtracting 9 from both sides of the equation.
24 - 9 = 9n + 9 - 9
This simplifies to:
15 = 9n
Step 3: Divide both sides by 9
Finally, we need to isolate n by dividing both sides of the equation by 9.
\frac{15}{9} = \frac{9n}{9}
This simplifies to:
\frac{5}{3} = n
Therefore, the value of n is \frac{5}{3}.
Solving the Second Equation: \frac{7}{10}c - 8 - \frac{1}{2}c = -16
The second equation is \frac{7}{10}c - 8 - \frac{1}{2}c = -16. To solve for c, we need to isolate the variable c on one side of the equation.
Step 1: Combine like terms
The first step is to combine the like terms on the left-hand side of the equation. In this case, we have two terms involving c, which are \frac{7}{10}c and -\frac{1}{2}c. We can combine these terms by finding a common denominator and adding their coefficients.
\frac{7}{10}c - \frac{1}{2}c = \frac{7}{10}c - \frac{5}{10}c
This simplifies to:
\frac{2}{10}c = \frac{2}{10}c
So, the equation becomes:
\frac{2}{10}c - 8 = -16
Step 2: Add 8 to both sides
Next, we need to get rid of the constant term on the left-hand side of the equation. We can do this by adding 8 to both sides of the equation.
\frac{2}{10}c - 8 + 8 = -16 + 8
This simplifies to:
\frac{2}{10}c = -8
Step 3: Multiply both sides by 10
Finally, we need to isolate c by multiplying both sides of the equation by 10.
10 \times \frac{2}{10}c = 10 \times -8
This simplifies to:
2c = -80
Step 4: Divide both sides by 2
Therefore, the value of c is -80/2, which simplifies to:
c = -40
Therefore, the value of c is -40.
Conclusion
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable on one side of the equation. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Q: What is the order of operations when solving a linear equation?
A: When solving a linear equation, you should follow the order of operations (PEMDAS):
- Parentheses: Evaluate any expressions inside parentheses.
- Exponents: Evaluate any exponential expressions.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.
Q: How do I handle fractions when solving a linear equation?
A: When solving a linear equation that involves fractions, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2.
Q: Can I use algebraic properties to solve linear equations?
A: Yes, you can use algebraic properties such as the distributive property, the commutative property, and the associative property to solve linear equations.
Q: How do I check my solution to a linear equation?
A: To check your solution to a linear equation, you can plug the value of the variable back into the original equation and see if it is true.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not following the order of operations
- Not isolating the variable on one side of the equation
- Not checking the solution
- Not using algebraic properties to simplify the equation
Q: Can I use technology to solve linear equations?
A: Yes, you can use technology such as calculators or computer software to solve linear equations. However, it is still important to understand the steps involved in solving linear equations and to be able to check your solution.
Q: How do I apply linear equations to real-world problems?
A: Linear equations can be applied to a wide range of real-world problems, such as:
- Modeling population growth
- Calculating the cost of goods
- Determining the amount of time it takes to complete a task
- Solving problems involving distance, rate, and time
By understanding how to solve linear equations and applying them to real-world problems, you can develop problem-solving skills and make informed decisions in a variety of contexts.