Problem 37 - Thinking Mathematically

Question

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(2 x+y=3\)

Step-by-Step Solution

Answer

The slope-intercept form of the line is \(y = -2x + 3\). The slope is -2, and the y-intercept is 3.

Step 1: Convert to slope-intercept form

To convert the equation \(2 x+y=3\) into slope-intercept form, isolate \(y\). Subtract \(2x\) from both sides of the equation to get \(y=-2x+3\)

Step 2: Identify the slope and y-intercept

In the slope-intercept form of the line equation, \(y = mx + c\), \(m\) is the slope and \(c\) is the y-intercept. Therefore the slope of the line is -2, and the y-intercept is 3.

Step 3: Graph the line

Start at the y-intercept 3 on the y-axis. From there, as the slope is negative which means going downward, move 2 units down for every 1 unit of movement to the right.

Key Concepts

Understanding the Slope

A slope represents how steep a line is. It tells you how the line moves as you go from left to right on a graph. In mathematical terms, slope is the ratio of the change in the y-values over the change in the x-values between any two points on a line. You might see this as "rise over run."
Here’s a simple way to remember:

• If the slope is positive, the line moves up as you travel from left to right.
• If the slope is negative, it moves down.
• If the slope is zero, the line is flat – horizontal.
• An undefined slope means the line is vertical.

In our example, the slope is -2. This means the line moves downward, falling 2 units for every 1 unit you move to the right. The steeper the slope (positive or negative), the more the line moves up or down.

Delving into the Y-intercept

The y-intercept is where the line crosses the y-axis. At this point, the x-value is zero. In the slope-intercept form of a line equation, expressed as \(y = mx + c\), the y-intercept is indicated by \(c\).
For the equation \(y = -2x + 3\), the y-intercept is the number 3. This means the line will cross the y-axis at the point \((0, 3)\).
Understanding the y-intercept helps ensure you start graphing from the correct point on the y-axis, setting the foundation for accurately portraying the entire line.

Graphing Equations Efficiently

Graphing equations becomes straightforward when you utilize the slope and y-intercept. First, always plot the y-intercept on the graph. In our example, place a point at \((0, 3)\).
Next, use the slope to determine the direction of your next point. With a slope of -2, move down 2 units and 1 unit to the right from the y-intercept. Place your next point there.
Draw a straight line through these points, extending it across the graph.
Tips for successful graphing:

• Check at least two points to ensure accuracy.
• Consider using graph paper for precise plotting.
• Label the axes and tick marks for clarity.

Having these steps in your toolkit makes graphing less daunting and more of an exploration.

Linear Equations Demystified

Linear equations are algebraic equations where each term is either a constant or the product of a constant and a single variable. When graphed, they produce a straight line.
They can be written in several forms, including standard form \(Ax + By = C\) and slope-intercept form \(y = mx + c\).
The slope-intercept form \(y = mx + c\) is often preferred for graphing because it clearly displays the slope \(m\) and y-intercept \(c\). Understanding these forms and how to manipulate them is crucial for solving and graphing linear equations.
Key features of linear equations include:

• They have constant rates of change, indicated by their slope.
• Their graphs are always straight lines.
• Their solutions involve finding x-values for specific y-values, and vice versa.

Grasping these fundamentals opens doors to more advanced mathematical concepts and real-life applications.