Problem 83 - 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 twice the \(x\) -variable by 5

Step-by-Step Solution

Answer

Translating the sentence into a mathematical equation gives \(y = 2x + 5\). When graphed, this equation presents a straight line with a slope of 2, passing through the y-axis at 5.

Step 1: Translate sentence to equation

Looking at the provided sentence, 'The \(y\) -variable exceeds twice the \(x\) -variable by 5', we can translate this into a mathematical equation. When we say \(y\) 'exceeds' something, it means that \(y\) is equal to that something plus a certain amount. Here, it means that \(y\) is equal to twice the \(x\)-variable plus 5. So this translates to the equation, \(y = 2x + 5\)

Step 2: Analyze equation

From the equation \(y = 2x + 5\), we can see that it is in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. In this case, the slope \(m = 2\), which means for every 1 unit increase in \(x\), \(y\) increases by 2 units. The y-intercept \(b = 5\) indicates that when \(x\) is 0, \(y\) is 5.

Step 3: Graph equation

Start by plotting the y-intercept (0, 5) on the graph. From this point, use the slope to find additional points and draw the line. Since slope \(m = 2\), which can be expressed as a fraction as \(m = 2/1\), for every 1 unit moved to the right along the x-axis, move 2 units up along the y-axis. Do this several times to find additional points and then draw a straight line connecting these points. Place arrows on both ends of the line to indicate it extends indefinitely. Your graph should show a line passing through points (0, 5), (1, 7), (2, 9) and so on.

Key Concepts

Slope-Intercept Form

Linear equations are often expressed using slope-intercept form because it provides an intuitive way to understand how changes in the equation affect the graph. The slope-intercept form is represented as \( y = mx + b \). Here, \( m \) is the slope of the line, and \( b \) is the y-intercept.- The **slope (**\( m \) ) describes the steepness or tilt of the line. It tells us how much \( y \) changes for a one-unit increase in \( x \).- The **y-intercept (**\( b \) ) is the point where the line crosses the y-axis. It's the value of \( y \) when \( x = 0 \).In our exercise, the equation is \( y = 2x + 5 \). The slope \( m \) is 2, which means for every unit increase in \( x \), \( y \) increases by two units. Meanwhile, the y-intercept \( b \) is 5, indicating the point (0,5). Understanding these parts makes it easier to visualize and graph the equation.

Graphing Equations

Graphing linear equations involves plotting points and using the slope to accurately draw the line. Start with the y-intercept, the point where the line crosses the y-axis. For our equation \( y = 2x + 5 \), the y-intercept is (0,5). Place this point on the y-axis of your graph.Next, use the slope to determine additional points. The slope \( m = 2 / 1 \) suggests that for every 1 unit you move right on the x-axis, move 2 units up along the y-axis.

• From (0, 5), move to (1, 7).
• Then to (2, 9), and so on.

Draw the line through these points and extend it in both directions with arrows, indicating it goes on forever. This graphical representation helps in visualizing how \( y \) depends on \( x \).

Two-Variable Equations

A two-variable linear equation, like \( y = 2x + 5 \), involves two variables where one depends on the other. The equation describes a relationship between \( x \) and \( y \) on a two-dimensional plane.The variables are typically called the independent variable \( x \) and the dependent variable \( y \):

• **Independent Variable (x):** This variable is free to change any value and doesn't depend on other variables.
• **Dependent Variable (y):** This is affected by the changes in the independent variable and is calculated using the equation.

For such equations like \( y = 2x + 5 \), if you choose different values for \( x \), you can find corresponding values for \( y \) to form coordinate pairs like (0, 5), (1, 7), etc., which can be plotted onto a graph.These types of equations are key in understanding how variables interact and are foundational in algebra.