Problem 32 - A Graphical Approach to College Algebra

Question

Graph each line by hand. Give the \(x\)- and y-intercepts. \(4 x-3 y=9\)

Step-by-Step Solution

Answer

x-intercept: \(\left(\frac{9}{4}, 0\right)\), y-intercept: \((0, -3)\).

Step 1: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Given \(4x - 3y = 9\), substitute \(x = 0\):\[4(0) - 3y = 9 \-3y = 9 \y = -3\]Thus, the y-intercept is \((0, -3)\).

Step 2: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). Again, start with \(4x - 3y = 9\), substitute \(y = 0\):\[4x - 3(0) = 9 \4x = 9 \x = \frac{9}{4}\]So, the x-intercept is \(\left(\frac{9}{4}, 0\right)\).

Step 3: Plot the intercepts on the Cartesian plane

On a graph, plot the points \((0, -3)\) for the y-intercept and \(\left(\frac{9}{4}, 0\right)\) for the x-intercept. These are the points where the line will cross the axes.

Step 4: Draw the line

Use a ruler to connect the two points, \((0, -3)\) and \(\left(\frac{9}{4}, 0\right)\), with a straight line. Extend the line beyond the intercepts.

Key Concepts

x-intercept

The x-intercept of a linear equation is the point where the line crosses the x-axis on a graph. At this point, the value of y is always zero because the line is not rising or falling vertically. To find the x-intercept from a given linear equation, you simply set y equal to zero and solve for x. For example, in the equation \( 4x - 3y = 9 \), substitute \( y = 0 \), and solve: \( 4x - 3(0) = 9 \) \( 4x = 9 \) \( x = \frac{9}{4} \) This gives the x-intercept as \( \left( \frac{9}{4}, 0 \right) \). Understanding where a line crosses the x-axis helps in visualizing the behavior of the graph, especially when sketching or analyzing its slope and direction.

y-intercept

The y-intercept of a line is the point where the line meets the y-axis. At this point, the value of x is zero. This simplifies the calculation since you only need to solve for one variable in the linear equation. For the given equation, \( 4x - 3y = 9 \), you find the y-intercept by setting \( x = 0 \) and solving for y: \( 4(0) - 3y = 9 \) \( -3y = 9 \) \( y = -3 \) Thus, the y-intercept is \( (0, -3) \). Locating the y-intercept aids in accurately plotting linear equations on a graph. It's a key point in exhibiting how the line behaves relative to the vertical axis.

Cartesian plane

The Cartesian plane, also known as the coordinate plane, is a two-dimensional surface where each point is defined by a pair of numerical coordinates. These coordinates are written as (x, y), representing positions relative to the two axes: x-axis (horizontal) and y-axis (vertical).

• The x-axis runs left and right, while the y-axis runs up and down.
• Intercepts lie on these axes and guide where the line will penetrate the grid.

When graphing linear equations, it's essential to accurately mark the intercepts on this plane. For the equation \( 4x - 3y = 9 \), plot the points \( (0, -3) \) for the y-intercept, and \( \left( \frac{9}{4}, 0 \right) \) for the x-intercept. Connecting these points with a straight line represents the solution graphically, depicting where the line traverses the plane.

linear equations

Linear equations express relationships where the highest power of any variable involved is one. These systems model straight-line graphs on a Cartesian plane. A standard form for linear equations is \( Ax + By = C \), where A, B, and C are constants.

• Linear equations yield straight lines when graphed.
• The slope of the line indicates its direction and steepness.
• Intercepts provide valuable anchor points for constructing the graph.

For example, our equation \( 4x - 3y = 9 \) represents such a line. By computing intercepts, you can draw this line on the graph, helping to visualize solutions and interpret data straightforwardly. Mastery of these concepts enables efficient analysis and understanding of linear relationships.