Problem 60 - College Algebra with Modeling and Visualization

Question

Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Graph the equation. \(15 x-y=30\)

Step-by-Step Solution

Answer

The x-intercept is at (2, 0) and the y-intercept is at (0, -30).

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). Start by substituting \(y = 0\) into the equation: \(15x - 0 = 30\). This simplifies to \(15x = 30\). Now, solve for \(x\) by dividing both sides by 15: \(x = \frac{30}{15} = 2\). So, the x-intercept is at the point (2, 0).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Substitute \(x = 0\) into the equation: \(15(0) - y = 30\). This simplifies to \(-y = 30\), so we solve for \(y\) by multiplying both sides by -1 to get \(y = -30\). Therefore, the y-intercept is at the point (0, -30).

Step 3: Plot Intercepts on the Graph

On a coordinate plane, plot the two points obtained from the intercepts: (2, 0) for the x-intercept and (0, -30) for the y-intercept. These will be the two points through which the line will pass.

Step 4: Draw the Line

Using a ruler, draw a straight line through the two points (2, 0) and (0, -30) on the graph. This line represents the graph of the equation \(15x - y = 30\).

Key Concepts

X-Intercept

The x-intercept is a key concept in graphing linear equations. It is the point where the graph of the equation crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept of an equation like \(15x - y = 30\), you set \(y = 0\) and solve for \(x\).

Here, substituting \(y = 0\) gives \(15x - 0 = 30\), which simplifies to \(15x = 30\). We solve for \(x\) by dividing both sides by 15, resulting in \(x = 2\). Therefore, the x-intercept is the point \((2, 0)\).

• Set \(y = 0\) to find x-intercept.
• Solve for \(x\).
• The x-intercept is where the graph crosses the x-axis.

Understanding the x-intercept helps to locate one of the key points needed to graph a linear equation.

Y-Intercept

The y-intercept is the point where a graph crosses the y-axis of a coordinate plane. At this point, the value of x is always zero. Finding the y-intercept is essential in plotting linear equations, as it gives a starting point on the vertical (y) axis.

For the given equation \(15x - y = 30\), set \(x = 0\) to find the y-intercept. Substituting \(x = 0\) into the equation gives \(15(0) - y = 30\), which simplifies to \(-y = 30\). Solving for \(y\), by multiplying both sides by -1, gives \(y = -30\). Thus, the y-intercept is the point \((0, -30)\).

• Set \(x = 0\) to find y-intercept.
• Solve for \(y\).
• The y-intercept is where the graph crosses the y-axis.

Accurately finding the y-intercept is key in graphing a straight line.

Coordinate Plane

A coordinate plane is a two-dimensional surface formed by two number lines intersecting at a right angle. These lines are called the axes. The horizontal axis is known as the x-axis, and the vertical one is the y-axis. The point where these axes intersect is called the origin, represented by \((0,0)\).

Points on the coordinate plane have an ordered pair \((x,y)\), which indicates their exact location. The x-coordinate tells how far to move horizontally from the origin, while the y-coordinate indicates how far to move vertically.

Plotting points on the coordinate plane is crucial when graphing equations. For instance, when graphing \(15x - y = 30\), you plot points such as \((2,0)\) and \((0,-30)\), which are the x- and y-intercepts.

• The x-axis is the horizontal line.
• The y-axis is the vertical line.
• The origin is the point \((0,0)\).
• Coordinates are written in the form \((x,y)\).

The coordinate plane makes it easier to visualize and solve linear equations.

Linear Equation

A linear equation is an equation between two variables that produces a straight line when graphed on a coordinate plane. The most general form of a linear equation in two variables is \(Ax + By + C = 0\), where A, B, and C are constants. However, it can also appear as \(y = mx + b\), known as the slope-intercept form, where \(m\) is the slope, and \(b\) is the y-intercept. In our specific case, the equation \(15x - y = 30\) is already in a form ready for graphing. By determining where the graph intersects the axes, we can plot the straight line. Linear equations are characterized by their consistent slope, which remains the same between any two points on the line.

These equations are simple yet powerful as they model many real-world phenomena.

• The general form is \(Ax + By + C = 0\).
• The slope-intercept form is \(y = mx + b\).
• Linear equations always produce straight lines on a graph.

Understanding linear equations is fundamental for graphing and interpreting data in mathematics.