Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write a linear equation that has the given solution. (There are many correct answers.) $$x=-2.5$$

Step-by-Step Solution

Answer

One possible linear equation that has the solution x = -2.5 is \(2x + 5 = 0\).

Step 1: Identify solution

Identify the given solution, which is x = -2.5 in this case.

Step 2: Generate a linear equation

As there can be any number of linear equations that have the solution x = -2.5, one possible linear equation can be generated by choosing a reasonable value for A and finding a corresponding value for B. For this solution, let's start by assuming A = 2. In this case, the equation will be 2x + B = 0. Substituting x = -2.5 into the equation gives 2*(-2.5) + B = 0 which simplifies to -5 + B = 0.

Step 3: Solve for B

Solve the equation -5 + B = 0 for B. Adding 5 to each side of the equation gives B = 5.

Step 4: Write the final linear equation

Replace B with the value found in step 3 to get the final linear equation which is 2x + 5 = 0.

Key Concepts

Linear Equation Solutions

Understanding linear equation solutions is crucial for students delving into algebra. A linear equation is, essentially, an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations typically take on the form of Ax + B = 0, where A and B are constants and x represents the variable.

When given a specific solution, like x = -2.5, our task is to determine a linear equation that resolves to this value. This implies that when x is substituted into this linear equation, the expression will simplify to zero. It's crucial to recognize that there are infinitely many linear equations that can satisfy a single solution. This concept is akin to the idea that there are numerous routes to reach a single destination.

When tasked with coming up with a linear equation for a given solution, you can start by selecting a suitable value for either coefficient A or the constant term B and then determine the other based on the provided solution. This process aligns with our understanding that a solution to a linear equation is essentially the value of the variable that makes the equation true.

Solving for Variables

Solving for variables is at the heart of algebra and a core skill in mathematics. When you have an equation, and you're asked to solve for a specific variable, this typically means you need to isolate the variable on one side of the equation. This process involves performing operations that will 'undo' the operations in the equation.

Applying Inverse Operations

In the context of our example, where we have the partial equation 2x + B = 0 and a given solution x = -2.5, our goal is to find the value of B. It involves applying inverse operations; in this case, we substitute x with -2.5 and then perform an addition or subtraction to isolate B.

It's important to remember to apply the same operation to both sides of the equation to maintain balance—what you do on one side, you do on the other. When students understand this foundation, they can approach solving for variables systematically for any value that a variable assumes. Ensuring that these operations are performed correctly and consistently is crucial for the accurate determination of variable values.

Generating Linear Equations

When generating linear equations, students wield creative freedom paired with logical constraints. A linear equation can be crafted in various ways as long as it adheres to the basic form of Ax + B = C, where A, B, and C are constants and x represents the variable. The task becomes interesting and slightly more complex when one is given a particular solution for x.

For instance, if x = -2.5, our job is to build an equation that when x is substituted, the equation holds true. We start by choosing a value for A, termed the coefficient of x. This can be any non-zero number as the coefficient determines the slope of the linear graph associated with the equation. Next, we solve for B to ensure that when x = -2.5 is plugged into the equation, the overall value does equate to zero, leading to the correct solution.

The creation of linear equations is not just mechanical but also logical, requiring sound reasoning to comply with the given conditions. The ability to generate a linear equation that corresponds to a specific solution underscores a deep comprehension of algebraic relationships and their graphical representations.