Problem 31 - Prealgebra

Question

The problems below review material we covered in Section 4.9 Graph each equation. $$x+y=3$$

Step-by-Step Solution

Answer

Graph the line by plotting points (3, 0), (0, 3), and (1, 2), and connect them in a straight line.

Step 1: Understand the Equation

The given equation is a linear equation representing a straight line: \( x + y = 3 \). This form \( ax + by = c \) can be rearranged to find points on the line.

Step 2: Find the Intercepts

To find the x-intercept, set \( y = 0 \) and solve for \( x \):\[ x + 0 = 3 \Rightarrow x = 3 \] The x-intercept is \( (3, 0) \).To find the y-intercept, set \( x = 0 \) and solve for \( y \):\[ 0 + y = 3 \Rightarrow y = 3 \] The y-intercept is \( (0, 3) \).

Step 3: Choose a Third Point

Select any other value for either \( x \) or \( y \) to find an additional point. Choose \( x = 1 \). Substitute into the equation:\[ 1 + y = 3 \Rightarrow y = 2 \]The third point is \( (1, 2) \).

Step 4: Plot the Points

Plot the points \((3,0), (0,3), (1,2)\) on a Cartesian coordinate plane.

Step 5: Draw the Line

Connect the plotted points with a straight line extending in both directions. This line represents the equation \( x + y = 3 \). Ensure the line passes through all the plotted points to confirm accuracy.

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear equation is the point where the line crosses the x-axis on a Cartesian coordinate plane. This happens when the value of y is zero. For any given linear equation in the format of \( ax + by = c \), finding the x-intercept is quite straightforward. Simply set \( y = 0 \), and solve for \( x \).

In our example, the equation is \( x + y = 3 \). To find the x-intercept, we set \( y = 0 \), which simplifies the equation to \( x + 0 = 3 \). Solving this gives \( x = 3 \). Therefore, the x-intercept is the point \( (3, 0) \).

• Key Point: The x-intercept is always on the x-axis, so the y-value is 0.
• Tip: Substitute \( y = 0 \) in the equation to find the x-intercept.

Exploring the Y-Intercept

The y-intercept of a line is where the line touches the y-axis. At this point, the x-value is zero. Finding the y-intercept involves setting \( x = 0 \) in the linear equation and solving for \( y \). In our example, the equation is \( x + y = 3 \). When we set \( x = 0 \), the equation simplifies to \( 0 + y = 3 \). Solving this gives \( y = 3 \). So, the y-intercept is \( (0, 3) \).

• Important Note: At the y-intercept, the x-value is always 0.
• Helpful Insight: It tells you where the line crosses the y-axis.

Navigating the Cartesian Coordinate Plane

The Cartesian coordinate plane is like a map used to plot points and draw graphs of equations. It has two axes: a horizontal axis called the x-axis, and a vertical axis called the y-axis. These axes divide the plane into four quadrants.

Points on this plane are identified by a pair of numbers \((x, y)\), known as coordinates. The first number represents the position along the x-axis, and the second number represents the position along the y-axis. To graph a linear equation, like \( x + y = 3 \), you need to find points like the x and y intercepts mentioned earlier. First, plot them on the plane.

• Visual Cue: Picture the plane as a flat grid where you can pinpoint exact locations using coordinates.
• Easy Graphing: Points such as x-intercept \((3, 0)\) and y-intercept \((0, 3)\) are essential for drawing a line that represents the equation.