Problem 82 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each sentence as a linear equation in two variables. Then graph the equation. The \(y\) -variable exceeds the \(x\) -variable by 4

Step-by-Step Solution

Answer

The linear equation corresponding to the given sentence is \(y = x + 4\). For graph, select few distinct values for \(x\) and calculate \(y\) values, then plot these pairs in a graph.

Step 1: Translate Sentence to Equation

When the problem states that the \(y\)-variable exceeds the \(x\)-variable by 4, it can be translated into the equation \(y = x + 4\). This means that for any given value of \(x\), \(y\) will be that value plus 4.

Step 2: Graphing the equation

To graph the equation \(y = x + 4\), start by choosing a few values for \(x\) and then calculate the corresponding value for \(y\) by adding 4. For example, if \(x = 0\), then \(y = 0 + 4 = 4\). If \(x = 1\), then \(y = 1 + 4 = 5\), and so forth. Plot these point pairs in a graph and draw a line through them to graph the complete equation.

Key Concepts

Graphing Linear Equations

Graphing linear equations helps us understand the relationship between two variables visually. To graph the given equation, you need points that follow the rule in the equation. Here is a simple way to achieve that:

• Start by picking any value for the x-variable, often 0 is a good start because calculations are easier.
• For each chosen x-value, calculate the corresponding y-value using the equation.
• In the equation \( y = x + 4 \), if you pick \( x = 0 \), \( y = 4 \), giving you the point (0, 4).
• Pick another value like \( x = 2 \), which results in \( y = 6 \), the point (2, 6).
• Plot these points on a graph. The x-axis represents x-values, and the y-axis is for y-values.

Once you have a few points, draw a straight line through them. This line is the graph of your linear equation. The slope of the line tells you how y changes with x in a consistent manner.

Two-Variable Equations

Two-variable equations involve relationships between two different variables. In this case, let's consider our equation \( y = x + 4 \):

• Here, "x" and "y" are variables. They can be any number, but they change depending on each other.
• The given equation shows that y is always 4 units more than x.

These types of equations can be expressed in the form \( y = mx + b \), which is called "slope-intercept form":

• m represents the slope of the line, showing the rate at which y increases or decreases as x does.
• b represents the y-intercept. It's the value of y when x is zero. Here, b = 4, meaning the line crosses the y-axis at y=4.

By understanding how these equations connect x and y, you can determine how changing one variable affects the other. This is useful in many real-world situations, like predicting outcomes or researching patterns.

Solving Equations

Solving linear equations is about finding values for variables that make the equation true. In the context of the exercise, solving isn't about finding an "answer" in the traditional sense, but about understanding the relationship between x and y:

• When you write \( y = x + 4 \), you effectively "solve" y in terms of x.
• Solving means that for any value of x, you can calculate y. This connects closely to graphing, where you visualize these solutions.
• Practically, you can input any number into the equation to find a corresponding y, which by graphing, helps illustrate the equation's solutions.

These kinds of solutions provide you not with one value, but with an entire set of points that form a straight line. All these points satisfy the equation. Learning to solve equations by understanding their form and solutions helps immensely in mastering linear relationships.