Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write a linear equation that has the given solution. (There are many correct answers.) $$x=\frac{1}{4}$$

Step-by-Step Solution

Answer

The linear equation that fulfills the problem's requirements is \(4x = 1\).

Step 1: Define the equation

Choose a value for a and b in the equation \(ax = b\), ensuring that when solving for x, the result will be \(x = \frac{1}{4}\). For simplicity, choose a = 4 and b = 1, which makes the equation \(4x = 1\).

Step 2: Solve for x

To confirm that the created equation is correct, solve for x. Doing so provides \(x = \frac{1}{4}\).

Key Concepts

Linear Equation Solutions

Understanding linear equation solutions is a foundational aspect of algebra that allows students to find the values that satisfy a given equation. When presented with a linear equation such as \( ax + b = 0 \), the goal is to determine the value of \( x \) that makes the equation true.

In our example, students are asked to write their own linear equation that yields the solution \( x = \frac{1}{4} \). The beauty of linear equations is their infinite possibilities for solutions; there's not just a single correct answer! By selecting appropriate values for \( a \) and \( b \) in the standard linear equation form, we can create a variety of equations that all have the same solution.

For instance, if we start with the equation \( 4x = 1 \), dividing both sides by 4 gives us the solution \( x = \frac{1}{4} \), which matches our target. But, we could equally write \( 8x = 2 \) or \( 12x = 3 \), and after the same type of simplification, find that \( x = \frac{1}{4} \) still holds true. This freedom demonstrates the infinite number of linear equations that can serve as correct representations for a given solution.

Solving Linear Equations

Solving linear equations is one of the most important skills in algebra. It involves manipulating the equation by performing the same operation on both sides to isolate the variable and find its value.

To illustrate, let's take the equation we derived earlier: \( 4x = 1 \). The solution process involves dividing both sides of the equation by the same non-zero number, which in this case is 4, to find the value of \( x \) that satisfies the equation. Once we do that, we end up with \( x = \frac{1}{4} \), confirming that the equation works.

Checking Your Solution

It's always good practice to check if your solution is accurate. For a linear equation like \( 4x = 1 \), substituting \( x \) with \( \frac{1}{4} \) should result in a true statement. In this example, \( 4 \times \frac{1}{4} = 1 \) verifies our solution. This step not only confirms the validity of the solution but also reinforces students' understanding of the properties of equality and operations carried out on equations.

Algebraic Equations

Algebraic equations are equations involving variables, constants, and arithmetic operations. They are used to represent relationships between quantities and are essential tools for solving a wide range of problems in mathematics. Linear equations are just one type of algebraic equation, characterized by each term being at most of the first degree.

The general form of a linear equation in one variable is \( ax + b = 0 \), where \( a \) and \( b \) are constants, and \( x \) is the variable we aim to solve for. It's important to understand that \( a \) cannot be zero since it would make the equation no longer linear.

Variations in Form

These equations can appear in different forms, such as \( ax + b = c \) or \( p(x) = q(x) \) where \( p \) and \( q \) are linear functions. Regardless of how they are presented, the approach to solving them is consistently rooted in the fundamental properties of algebra, like the distributive property, the associative property, and the commutative property. Mastery of solving algebraic equations paves the way for understanding more complex mathematical concepts and solving real-world problems.