The Property Used in Step 1

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The Linear Equation: A Step-by-Step Solution

In mathematics, linear equations are a fundamental concept that plays a crucial role in various branches of mathematics, science, and engineering. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will explore the solution of a linear equation using a step-by-step approach.

The Given Linear Equation

The given linear equation is:

$\frac{1}{3}(12x+15)=7$

This equation can be solved using the distributive property, which states that for any real numbers a, b, and c, the following equation holds:

$a(b+c) = ab + ac$

Using this property, we can simplify the given equation.

Step 1: Simplifying the Equation

To simplify the equation, we need to multiply both sides of the equation by 3 to eliminate the fraction.

$12x+15=21$

Next, we need to isolate the term containing the variable x. To do this, we can subtract 15 from both sides of the equation.

$12x=6$

Step 2: Isolating the Variable

Now, we need to isolate the variable x. To do this, we can divide both sides of the equation by 12.

$x=\frac{6}{12}$

Simplifying the fraction, we get:

$x=\frac{1}{2}$

The Property Used in Step 1

The property used in step 1 was the distributive property, which states that for any real numbers a, b, and c, the following equation holds:

$a(b+c) = ab + ac$

This property was used to simplify the equation by multiplying both sides of the equation by 3.

The Importance of Linear Equations

Linear equations are an essential concept in mathematics, science, and engineering. They are used to model real-world problems, such as the motion of objects, the flow of fluids, and the growth of populations. Linear equations are also used in various fields, such as economics, physics, and computer science.

Solving Linear Equations

Solving linear equations involves a series of steps, including simplifying the equation, isolating the variable, and solving for the variable. The steps involved in solving a linear equation are:

Simplifying the equation by combining like terms.
Isolating the variable by adding or subtracting the same value to both sides of the equation.
Solving for the variable by dividing both sides of the equation by the coefficient of the variable.

Real-World Applications of Linear Equations

Linear equations have numerous real-world applications, including:

Motion of Objects: Linear equations are used to model the motion of objects, such as the trajectory of a projectile or the motion of a car.
Flow of Fluids: Linear equations are used to model the flow of fluids, such as the flow of water through a pipe or the flow of air through a duct.
Growth of Populations: Linear equations are used to model the growth of populations, such as the growth of a population of bacteria or the growth of a population of animals.
Economics: Linear equations are used in economics to model the behavior of economic systems, such as the supply and demand of a product.
Physics: Linear equations are used in physics to model the behavior of physical systems, such as the motion of a pendulum or the vibration of a spring.

Conclusion

In conclusion, linear equations are a fundamental concept in mathematics, science, and engineering. They are used to model real-world problems and have numerous real-world applications. Solving linear equations involves a series of steps, including simplifying the equation, isolating the variable, and solving for the variable. By understanding linear equations, we can better understand the world around us and make more informed decisions.

References

[1] "Linear Equations" by Math Open Reference
[2] "Solving Linear Equations" by Khan Academy
[3] "Linear Equations in Real-World Applications" by Wolfram MathWorld

Further Reading

"Linear Algebra" by Gilbert Strang
"Introduction to Linear Equations" by Michael Artin
"Linear Equations and Inequalities" by James R. Smith
Frequently Asked Questions (FAQs) about Linear Equations

In this article, we will answer some of the most frequently asked questions about 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. It is a simple equation that can be solved using basic algebraic operations.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

Simplify the equation by combining like terms.
Isolate the variable by adding or subtracting the same value to both sides of the equation.
Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers a, b, and c, the following equation holds:

$a(b+c) = ab + ac$

This property is used to simplify equations by multiplying both sides of the equation by a constant.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, in the equation $2x + 3x$ , the like terms are $2x$ and $3x$ . You can combine these terms by adding them together.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, you need to add or subtract the same value to both sides of the equation. For example, in the equation $x + 2 = 5$ , you can isolate the variable by subtracting 2 from both sides of the equation.

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. A quadratic equation, on the other hand, 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, while the equation $x + 2 = 5$ is a linear equation.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, you need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. For example, in the equation $\frac{1}{2}x + 2 = 3$ , you can multiply both sides of the equation by 2 to eliminate the fraction.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications, including:

Motion of Objects: Linear equations are used to model the motion of objects, such as the trajectory of a projectile or the motion of a car.
Flow of Fluids: Linear equations are used to model the flow of fluids, such as the flow of water through a pipe or the flow of air through a duct.
Growth of Populations: Linear equations are used to model the growth of populations, such as the growth of a population of bacteria or the growth of a population of animals.
Economics: Linear equations are used in economics to model the behavior of economic systems, such as the supply and demand of a product.
Physics: Linear equations are used in physics to model the behavior of physical systems, such as the motion of a pendulum or the vibration of a spring.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to plot two points on the graph and draw a line through them. You can find the x-intercept and y-intercept of the equation by setting the variable equal to zero and solving for the other variable.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is $y = mx + b$ , where m is the slope of the line and b is the y-intercept.

Q: How do I find the slope of a linear equation?

A: To find the slope of a linear equation, you need to divide the change in y by the change in x. For example, in the equation $y = 2x + 3$ , the slope is 2.

Q: What is the difference between a linear equation and a nonlinear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. A nonlinear equation, on the other hand, is an equation in which the highest power of the variable(s) is greater than 1. For example, the equation $x^2 + 2x + 1 = 0$ is a nonlinear equation, while the equation $x + 2 = 5$ is a linear equation.

Conclusion

References

[1] "Linear Equations" by Math Open Reference
[2] "Solving Linear Equations" by Khan Academy
[3] "Linear Equations in Real-World Applications" by Wolfram MathWorld

Further Reading

"Linear Algebra" by Gilbert Strang
"Introduction to Linear Equations" by Michael Artin
"Linear Equations and Inequalities" by James R. Smith