Problem 77 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The line \(2 y=3 x+7\) has a \(y\) -intercept of 7.

Step-by-Step Solution

Answer

The statement is false. The correct statement is 'The line \(2y = 3x + 7\) has a y-intercept of 3.5'.

Step 1: Write the Equation in Standard form

The given equation is \(2y = 3x + 7\). To bring it into the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, divide every term in the equation by 2 to get \(y = 1.5x + 3.5\). Now, it is in the standard form.

Step 2: Identify the y-intercept

The y-intercept is the value of \(y\) when \(x = 0\). Looking at the equation, we can see that when \(x = 0\), \(y = 3.5\). Therefore, the y-intercept is 3.5.

Step 3: Check the Answer

The statement is 'The line \(2y = 3x+7\) has a y-intercept of 7'. From step 2, we find that the actual y-intercept is 3.5. Hence, the statement is false.

Step 4: Correct the Statement

To make the given statement true, it should be corrected as 'The line \(2y = 3x+7\) has a y-intercept of 3.5'.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and represent straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables is written as \( ax + by = c \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are the variables.
Linear equations show how two variables relate to each other proportionally. For example, when one variable increases, the other might also increase, decrease, or stay the same based on the constants' values.
Some characteristics of linear equations include:

• They produce a straight line when plotted.
• The equation doesn't have exponents higher than one.
• The solutions are all the points on the line that satisfy the equation.

The fun part about these equations is their simplicity. They help in understanding the initial steps in solving more complex algebraic equations, laying the groundwork for further math topics.

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations using the formula \( y = mx + b \). Here, \( m \) denotes the slope of the line and \( b \) gives the y-intercept. This form is incredibly useful for quickly graphing a line or understanding its characteristics instantly.

• Slope (\( m \)): It represents how steep the line is. A higher slope means a steeper line, whereas a slope of zero indicates a flat line parallel to the x-axis.
• Y-intercept (\( b \)): This is where the line crosses the y-axis. It provides the starting value of \( y \) when \( x \) is zero.

The slope-intercept form is preferable because it directly showcases the key attributes of the line, making it easy for students to visualize and interpret the graph. It's a great tool in algebra to transition from numerical solutions to visual comprehension.

Y-Intercept

The y-intercept of a line is a critical concept in understanding the line's behavior on a graph. Defined simply, the y-intercept is the point where the line crosses the y-axis. This occurs when the value of \( x \) is zero, highlighting the position of the line relative to the y-axis.
In the equation \( y = mx + b \), \( b \) is the y-intercept. For instance, in the equation \( y = 1.5x + 3.5 \), the y-intercept is 3.5. This means if you start at the origin and move directly up or down the y-axis, the point (0, 3.5) is where the line intersects.
Why is this important? Knowing the y-intercept provides:

• A starting point to graph the line (since you begin at the y-intercept and then use the slope to determine the direction and steepness of the line).
• It gives a quick understanding of the equation's real-world meaning, since it often represents the initial value in applied contexts, like the starting balance in a bank account or the baseline measurement in an experiment.

Understanding the y-intercept offers a strong foundation in graphically interpreting linear equations and predicting their behavior across different values of \( x \).