What Value Of $x$ Makes The Equation Below True?$4x + 7 = 27$
Introduction to Solving Linear Equations
In mathematics, solving linear equations is a fundamental concept that helps us find the value of unknown variables. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving a simple linear equation of the form $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable we want to solve for.
The Equation to Solve
The equation we want to solve is $4x + 7 = 27$. This equation is a linear equation in one variable, where the highest power of the variable $x$ is 1. Our goal is to find the value of $x$ that makes this equation true.
Step 1: Isolate the Variable Term
To solve for $x$, we need to isolate the variable term on one side of the equation. In this case, we can start by subtracting 7 from both sides of the equation. This will help us get rid of the constant term on the same side as the variable term.
Step 2: Simplify the Equation
After subtracting 7 from both sides, we get:
Step 3: Solve for $x$
Now that we have isolated the variable term, we can solve for $x$ by dividing both sides of the equation by 4.
Step 4: Simplify the Result
After dividing both sides by 4, we get:
Conclusion
In this article, we solved a simple linear equation of the form $ax + b = c$. We started by isolating the variable term on one side of the equation, then simplified the equation, and finally solved for $x$ by dividing both sides by the coefficient of the variable term. The value of $x$ that makes the equation true is $x = 5$.
Tips and Tricks for Solving Linear Equations
- Always start by isolating the variable term on one side of the equation.
- Use inverse operations to get rid of the constant term on the same side as the variable term.
- Simplify the equation as much as possible to make it easier to solve.
- Check your solution by plugging it back into the original equation.
Real-World Applications of Solving Linear Equations
Solving linear equations has many real-world applications, such as:
- Finding the cost of an item based on its price and the number of items purchased.
- Determining the amount of time it takes to complete a task based on the rate of work and the amount of work done.
- Calculating the interest on a loan based on the principal amount, interest rate, and time period.
Common Mistakes to Avoid When Solving Linear Equations
- Not isolating the variable term on one side of the equation.
- Not using inverse operations to get rid of the constant term on the same side as the variable term.
- Not simplifying the equation as much as possible.
- Not checking the solution by plugging it back into the original equation.
Final Thoughts
Solving linear equations is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve simple linear equations and apply the concepts to real-world problems. Remember to always isolate the variable term, use inverse operations, simplify the equation, and check your solution to ensure accuracy.
Introduction
Solving linear equations is a fundamental concept in mathematics that has many real-world applications. In this article, we will answer some frequently asked questions (FAQs) about solving linear equations.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. For example, the equation $2x + 3 = 5$ is a linear equation because the highest power of the variable $x$ is 1.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable term on one side of the equation. You can do this by using inverse operations, such as addition, subtraction, multiplication, and division. For example, to solve the equation $2x + 3 = 5$, you can subtract 3 from both sides to get $2x = 2$, and then divide both sides by 2 to get $x = 1$.
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. For example, the equation $x^2 + 2x + 1 = 0$ is a quadratic equation because the highest power of the variable $x$ is 2.
Q: Can I use a calculator to solve a linear equation?
A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your solution by plugging it back into the original equation to ensure accuracy.
Q: What if I have a linear equation with fractions?
A: If you have a linear equation with fractions, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions. For example, to solve the equation $\frac{2x}{3} + 1 = 2$, you can multiply both sides by 3 to get $2x + 3 = 6$, and then subtract 3 from both sides to get $2x = 3$, and finally divide both sides by 2 to get $x = \frac{3}{2}$.
Q: Can I solve a linear equation with variables on both sides?
A: Yes, you can solve a linear equation with variables on both sides. To do this, you need to isolate one of the variables on one side of the equation, and then solve for the other variable. For example, to solve the equation $2x + 3 = 5x - 2$, you can subtract 2x from both sides to get $3 = 3x - 2$, and then add 2 to both sides to get $5 = 3x$, and finally divide both sides by 3 to get $x = \frac{5}{3}$.
Q: What if I have a linear equation with decimals?
A: If you have a linear equation with decimals, you can multiply both sides of the equation by a power of 10 to eliminate the decimals. For example, to solve the equation $2.5x + 3 = 5.2$, you can multiply both sides by 10 to get $25x + 30 = 52$, and then subtract 30 from both sides to get $25x = 22$, and finally divide both sides by 25 to get $x = \frac{22}{25}$.
Q: Can I solve a linear equation with absolute values?
A: Yes, you can solve a linear equation with absolute values. To do this, you need to isolate the absolute value term on one side of the equation, and then solve for the variable. For example, to solve the equation $|2x - 3| = 5$, you can split the equation into two separate equations: $2x - 3 = 5$ and $2x - 3 = -5$, and then solve each equation separately.
Conclusion
Solving linear equations is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve simple linear equations and apply the concepts to real-world problems. Remember to always isolate the variable term, use inverse operations, simplify the equation, and check your solution to ensure accuracy.