Problem 31 - Algebra 1

Question

Rewrite the equation so that \(x\) is a function of \(y .\) Then use the result to find \(x\) when \(y=-2,-1,0,\) and 1. $$3 y-x=12$$

Step-by-Step Solution

Answer

The function is \(x = 3y - 12\). When \(y = -2\), \(x = -18\). When \(y = -1\), \(x = -15\). When \(y = 0\), \(x = -12\). When \(y = 1\), \(x = -9\).

Step 1: Rearrange the equation to make \(x\) the subject

In the equation \(3y - x = 12\), to isolate \(x\), we add \(x\) to both sides, and subtract \(12\) from both sides, resulting in \(x = 3y - 12\). This gives the function of \(x\) in terms of \(y\).

Step 2: Substitute \(y = -2\) into the equation

Using the function \(x = 3y - 12\), substitute \(-2\) for \(y\). This results in \(x = 3(-2) - 12 = -6 - 12 = -18\). So when \(y = -2\), \(x = -18\).

Step 3: Substitute \(y = -1\) into the equation

Substitute \(-1\) for \(y\) in the function \(x = 3y - 12\). This gives \(x = 3(-1) - 12 = -3 - 12 = -15\). So when \(y = -1\), \(x = -15\).

Step 4: Substitute \(y = 0\) into the equation

Substitute \(0\) for \(y\) in the function \(x = 3y - 12\). This gives \(x = 3(0) - 12 = 0 - 12 = -12\). So when \(y = 0\), \(x = -12\).

Step 5: Substitute \(y = 1\) into the equation

Substitute \(1\) for \(y\) in the function \(x = 3y - 12\). This gives \(x = 3(1) - 12 = 3 - 12 = -9\). So when \(y = 1\), \(x = -9\).

Key Concepts

Solving Equations and Function Notation

In the world of mathematics, solving equations is a fundamental skill. It involves finding the values of variables that satisfy a given equation. In other words, you want to discover what numbers make the equation true. For instance, consider the equation \(3y - x = 12\). The task here is to express \(x\) as a function of \(y\). This means manipulating the equation until \(x\) is alone on one side:

• First, add \(x\) to both sides to eliminate the negative sign.
• Then, subtract \(12\) from both sides to further isolate \(x\).
• You'll end up with \(x = 3y - 12\).

This simplified function, \(x = 3y - 12\), allows you to determine the value of \(x\) for any given value of \(y\). By rewriting the equation, you've now transformed it into a function notation, often written as \(x(y)\), to describe the relationship between \(x\) and \(y\). This makes it easier to substitute values and solve further.

Understanding Substitution in Equations

Substitution is a handy math technique that involves replacing variables in an equation with specific values. It’s much like filling in the blanks in a sentence. By substituting a given value for a variable, you can determine the result of an equation. Let’s break it down using our function \(x = 3y - 12\):

• If \(y = -2\), substitute to get \(x = 3(-2) - 12 = -18\).
• If \(y = -1\), substitute to get \(x = 3(-1) - 12 = -15\).
• If \(y = 0\), substitute to get \(x = 3(0) - 12 = -12\).
• If \(y = 1\), substitute to get \(x = 3(1) - 12 = -9\).

Substituting these specific values of \(y\) allows us to calculate what the corresponding \(x\) values will be. This technique is especially helpful for verifying the relationship described by the equation or function, ensuring that it holds true for all values substituted.

Variables Acting as Functions in Equations

In equations, we often encounter variables that can function as independent and dependent variables. This is like having one thing affect another. When a variable is the subject of a function, like \(x = 3y - 12\), \(x\) is dependent on \(y\). This means whenever \(y\) changes, \(x\) reacts and changes accordingly.

• **Independent Variable (\(y\)):** The value we choose or control. In this scenario, \(y\) is independent because we are deciding its value.
• **Dependent Variable (\(x\)):** The value that changes in response to the independent variable. Here, \(x\) depends on what \(y\) is.

Viewing \(x\) as a function of \(y\) helps us to understand the causal relationship between them. This understanding is not only critical in math but also incredibly useful in real-world contexts, where we often explore how changing one aspect influences another. Whether it's a business adjusting prices or a scientist altering a variable in an experiment, recognizing this function relationship is key.