Problem 47 - Algebra A Combined Function

Question

Write a single linear equation in three variables that has (-1,2,-4) as a solution. (There are many possibilities.) Explain the process you used to write an equation.

Step-by-Step Solution

Answer

One equation is \( x + 2y + z = -1 \).

Step 1: Understand the Components

A single linear equation with three variables is of the form \( ax + by + cz = d \), where \( a, b, c, \) and \( d \) are constants, and \( x, y, z \) are the variables. To have a specific solution, such as \((-1, 2, -4)\), means substituting these values into the equation should satisfy it.

Step 2: Substitute the Solution

Substitute \( x = -1 \), \( y = 2 \), and \( z = -4 \) into the general equation \( ax + by + cz = d \). This gives us the equation \( a(-1) + b(2) + c(-4) = d \).

Step 3: Assign Coefficients

Choose coefficients for \( a, b, \) and \( c \). For simplicity, let's choose \( a = 1, b = 2, c = 1 \).

Step 4: Substitute Coefficients and Solve for d

Plug in the chosen coefficients into the equation with the solution substitution: \( 1(-1) + 2(2) + 1(-4) = d \). Simplify this to find \( d \): \( -1 + 4 - 4 = d \). Thus, \( d = -1 \).

Step 5: Write the Final Equation

With the chosen coefficients and calculated \( d \), the equation becomes \( x + 2y + z = -1 \).

Key Concepts

Three Variables

When discussing linear equations involving three variables, we're typically referring to an equation of the form \( ax + by + cz = d \). In this setup, \(x\), \(y\), and \(z\) are the variables, and their behavior or interaction within the equation is determined by the constants \(a\), \(b\), \(c\), and \(d\). Each of these variables represents a dimension within a system, allowing the equation to extend into a three-dimensional space.
Understanding the concept of three variables is crucial because it allows us to explore more complex systems beyond just one or two dimensions, offering insights into multi-factor interactions.

Solution Substitution

Solution substitution is a technique used to verify a given solution within an equation. In the realm of linear equations with three variables, if you have a proposed solution like \(-1, 2, -4\), substitution involves plugging these values into the equation \( ax + by + cz = d \).
Here's how it works:

• Substitute \(x = -1\), \(y = 2\), and \(z = -4\) into the equation.
• Calculate the left side of the equation.
• Check if it equals the constant \(d\) on the right side.

This process ensures the solution is correct by checking its compatibility with the equation's structure.

Coefficients

Coefficients in a linear equation are the constants that multiply the variables. For an equation with three variables, these are \(a\), \(b\), and \(c\). The choice of these coefficients affects not only the equation's form but also its solution set and behavior.
When choosing coefficients:

• Simplicity can be key; pick manageable numbers to keep computations straightforward.
• The ratios between different coefficients can adjust the orientation of the solution plane in a three-dimensional system.

By altering the coefficients, you essentially define the features and directionality of the equation, impacting how solutions, like points or lines, behave or appear.

Solving Equations

To successfully solve linear equations, especially those with multiple variables, it is essential to break down the process into clear steps. For each specific equation, this involves:

• Identifying the equation format and variables: here, it's \( ax + by + cz = d \).
• Substituting known solutions into the equation to ensure their correctness.
• Finding suitable coefficients for variables to achieve a meaningful and valid equivalent of \(d\).

From these steps, solving the equation involves ensuring all chosen values satisfy the equation when substituted. This attention to each element in solving helps in confirming the correctness of the entire setup and the proposed solutions, ensuring they make logical sense within the presented equation.