Solve And Check The Linear Equation.${ 5x - (4x - 1) = 2 }$

Mar 7, 2025 by ADMIN 61 views

Solve and Check the Linear Equation

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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 and checking a linear equation of the form $ax + b = c$ , where $a$ , $b$ , and $c$ are constants. We will use the given equation $5x - (4x - 1) = 2$ as an example to demonstrate the step-by-step process of solving and checking a linear equation.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable (in this case, $x$ ) is 1. It can be written in the form $ax + b = c$ , where $a$ , $b$ , and $c$ are constants. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Given Equation

The given equation is $5x - (4x - 1) = 2$ . This equation can be solved using the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ .

Step 1: Apply the Distributive Property

To solve the equation, we first need to apply the distributive property to the term $-(4x - 1)$ . This will give us:

5x - (4x - 1) = 5x - 4x + 1

Step 2: Simplify the Equation

Now, we can simplify the equation by combining like terms:

5x - 4x + 1 = x + 1

Step 3: Isolate the Variable

Next, we need to isolate the variable $x$ by subtracting 1 from both sides of the equation:

x + 1 = 2

Step 4: Solve for x

Finally, we can solve for $x$ by subtracting 1 from both sides of the equation:

x = 1

Checking the Solution

To check our solution, we can plug $x = 1$ back into the original equation:

5(1) - (4(1) - 1) = 5 - 3 = 2

Since the equation is true, we have confirmed that our solution is correct.

Conclusion

In this article, we have demonstrated the step-by-step process of solving and checking a linear equation. We have used the given equation $5x - (4x - 1) = 2$ as an example to illustrate the use of the distributive property, simplification, and isolation of the variable. By following these steps, we have arrived at the solution $x = 1$ , which we have confirmed by plugging it back into the original equation.

Frequently Asked Questions

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the distributive property, simplify the equation, isolate the variable, and solve for the variable.

Q: How do I check my solution?

A: To check your solution, plug the value of the variable back into the original equation and verify that it is true.

Tips and Tricks

Tip 1: Use the distributive property to simplify the equation.

The distributive property can help you simplify the equation and make it easier to solve.

Tip 2: Isolate the variable by adding or subtracting the same value from both sides of the equation.

Isolating the variable can help you solve for the variable and check your solution.

Tip 3: Check your solution by plugging the value of the variable back into the original equation.

Checking your solution can help you verify that your answer is correct.

Real-World Applications

Linear equations have many real-world applications, including:

Physics: Linear equations are used to describe the motion of objects under constant acceleration.
Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, solving and checking linear equations is a crucial skill for students to master. By following the step-by-step process outlined in this article, you can solve and check linear equations with confidence. Remember to use the distributive property, simplify the equation, isolate the variable, and check your solution to ensure that your answer is correct.

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Frequently Asked Questions

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. It can be written in the form $ax + b = c$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the distributive property, simplify the equation, isolate the variable, and solve for 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 $c$ , $a(b + c) = ab + ac$ . This property can be used to simplify equations and make them easier to solve.

Q: How do I isolate the variable?

A: To isolate the variable, you can add or subtract the same value from both sides of the equation. This will help you get the variable by itself and solve for its value.

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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, $x + 2 = 3$ is a linear equation, while $x^2 + 2x + 1 = 0$ is a quadratic equation.

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 the value of the variable back into the original equation.

Q: How do I check my solution?

A: To check your solution, plug the value of the variable back into the original equation and verify that it is true.

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.

Q: Can I use linear equations to model real-world problems?

A: Yes, linear equations can be used to model real-world problems. For example, you can use linear equations to describe the motion of objects under constant acceleration, or to model the cost of producing a certain number of items.

Advanced Questions

Q: What is the concept of linear independence?

A: Linear independence is a concept in linear algebra that refers to the ability of a set of vectors to span a vector space. In other words, a set of vectors is linearly independent if none of the vectors can be expressed as a linear combination of the others.

Q: How do I determine if a set of vectors is linearly independent?

A: To determine if a set of vectors is linearly independent, you can use the concept of linear combinations. If the only way to express the zero vector as a linear combination of the vectors is with all coefficients equal to zero, then the vectors are linearly independent.

Q: What is the concept of linear transformation?

A: A linear transformation is a function between vector spaces that preserves the operations of vector addition and scalar multiplication. In other words, a linear transformation is a function that maps vectors to vectors in a way that preserves the properties of vector addition and scalar multiplication.

Q: How do I determine if a function is a linear transformation?

A: To determine if a function is a linear transformation, you can use the concept of linearity. If the function preserves the operations of vector addition and scalar multiplication, then it is a linear transformation.

Real-World Applications

Q: How are linear equations used in physics?

A: Linear equations are used in physics to describe the motion of objects under constant acceleration. For example, the equation $s = ut + \frac{1}{2}at^2$ is a linear equation that describes the position of an object as a function of time.

Q: How are linear equations used in engineering?

A: Linear equations are used in engineering to design and optimize systems, such as electrical circuits and mechanical systems. For example, the equation $V = IR$ is a linear equation that describes the relationship between voltage, current, and resistance in an electrical circuit.

Q: How are linear equations used in economics?

A: Linear equations are used in economics to model economic systems and make predictions about future trends. For example, the equation $C = a + bY$ is a linear equation that describes the relationship between consumption and income.

Conclusion

In conclusion, linear equations are a fundamental concept in mathematics that have many real-world applications. By understanding how to solve and check linear equations, you can apply this knowledge to a wide range of fields, including physics, engineering, and economics. Remember to use the distributive property, simplify the equation, isolate the variable, and check your solution to ensure that your answer is correct.