Problem 30 - A Graphical Approach to College Algebra

Question

Graph each line by hand. Give the \(x\)- and y-intercepts. \(2 x-3 y=6\)

Step-by-Step Solution

Answer

The x-intercept is (3, 0) and the y-intercept is (0, -2). Draw a line through these points.

Step 1: Identify the Equation

The equation given is a linear equation in standard form: \(2x - 3y = 6\). Our task is to graph this by hand and find the \(x\)- and \(y\)-intercepts.

Step 2: Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation and solve for \(x\).\[2x - 3(0) = 6 \2x = 6 \x = \frac{6}{2} = 3\]So, the \(x\)-intercept is \((3, 0)\).

Step 3: Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation and solve for \(y\).\[2(0) - 3y = 6 \-3y = 6 \y = \frac{6}{-3} = -2\]So, the \(y\)-intercept is \((0, -2)\).

Step 4: Graph the Line

Plot the \(x\)-intercept \((3, 0)\) and the \(y\)-intercept \((0, -2)\) on the graph. Draw a straight line through these two points to graph the equation.

Key Concepts

Graphing Lines

Graphing lines involves visualizing a linear equation on a coordinate plane. Linear equations, like the one given, form straight lines when graphed. The primary goal is to determine two key points: the \(x\)- and \(y\)-intercepts. These points allow us to draw the line accurately. To graph a line from its equation, follow these steps:

• Identify the equation: Determine its form (like \(ax + by = c\) which is the standard form).
• Find intercepts: Calculate where the line crosses the \(x\)- and \(y\)-axes (these points are crucial).
• Plot the intercepts: Place these points on the graph.
• Draw the line: Connect the dots with a ruler to extend the line through the intercepts.

By following this simple process, you can graph any linear equation effectively. The line you draw represents all the solutions (or points) that satisfy the equation.

x-intercept

The \(x\)-intercept of a line is the point where the line crosses the \(x\)-axis. At this point, the value of \(y\) is zero. To find the \(x\)-intercept from a linear equation, substitute \(y = 0\) into the equation and solve for \(x\). For the equation \(2x - 3y = 6\), this process looks like:\[2x - 3(0) = 6\]\[2x = 6\]\[x = \frac{6}{2} = 3\]Thus, the \(x\)-intercept is \((3, 0)\). This point, \((3, 0)\), tells us where the graph touches or crosses the \(x\)-axis. It is essential for plotting the graph, as it is one of the points that define the line.

y-intercept

The \(y\)-intercept is the point where the line crosses the \(y\)-axis. At the \(y\)-intercept, the value of \(x\) is always zero. To determine the \(y\)-intercept from a linear equation, substitute \(x = 0\) and solve for \(y\). In our example equation \(2x - 3y = 6\), this is calculated as follows:\[2(0) - 3y = 6\]\[-3y = 6\]\[y = \frac{6}{-3} = -2\]Thus, the \(y\)-intercept is \((0, -2)\). This point \((0, -2)\) is crucial for graphing the line, as it shows where the line meets the \(y\)-axis. Both the \(x\)- and \(y\)-intercepts are used to help draw the accurate graph of the line.