Problem 39 - Contemporary Precalculus

Question

Graph the equation. Label all intercepts. $$2 y-x=2$$

Step-by-Step Solution

Answer

Answer: The x-intercept of the equation is at the point (-2, 0) and the y-intercept is at the point (0, 1). To graph this equation, plot the intercepts on a coordinate plane, then draw a line passing through both intercepts which represents the graph for the equation $$2y - x = 2$$.

Step 1: Identify the equation

The given equation is: $$2y - x = 2$$

Step 2: Find the x-intercept

To find the x-intercept, set y = 0 in the given equation and solve for x: $$2(0) - x = 2 \Rightarrow x = -2$$ So, the x-intercept is at the point (-2, 0).

Step 3: Find the y-intercept

To find the y-intercept, set x = 0 in the given equation and solve for y: $$2y - 0= 2 \Rightarrow y = 1$$ So, the y-intercept is at the point (0, 1).

Step 4: Plot the intercepts and graph the equation

Using the x and y-intercepts (-2, 0) and (0, 1), plot the points on a coordinate plane. Draw a line passing through both intercepts, which represents the graph for the equation $$2y - x = 2$$. Label the intercepts.

Key Concepts

x-intercept

The x-intercept is an important feature when learning about linear equations. It is the point where the graph of a line crosses the x-axis on a coordinate plane. This means that at the x-intercept, the value of y is always zero.

To find the x-intercept, you simply substitute y = 0 into the equation and solve for x. For example, in the equation \(2y - x = 2\), by setting y to zero, the equation simplifies to \(-x = 2\), giving us \(x = -2\). Thus, the x-intercept is the point \((-2, 0)\).

Understanding how to find and interpret the x-intercept helps in sketching the graph and understanding the behavior of the line in relation to the x-axis.

y-intercept

The y-intercept is another key element of linear equations. It is the point on the graph where the line crosses the y-axis. At this point, the value of x is zero.

To find the y-intercept, set x to zero in the equation and solve for y. For example, with the equation \(2y - x = 2\), setting \(x = 0\) gives \(2y = 2\), leading to \(y = 1\). Hence, the y-intercept occurs at the point \((0, 1)\).

The y-intercept provides a starting point for graphing the line and helps in understanding how the graph behaves vertically.

coordinate plane

A coordinate plane is a two-dimensional surface where we can graph equations like \(2y - x = 2\). It contains two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).

The intersection point of these two axes is known as the origin, or \((0, 0)\). The coordinate plane helps to visualize the relationship between variables in an equation by plotting points that correspond to x and y values.

• Each point on the plane is represented as \((x, y)\), where x is the horizontal distance from the origin, and y is the vertical distance.
• Drawing axes allows us to graphically represent equations and understand their intercepts at a glance.

Understanding the coordinate plane is fundamental to graphing and interpreting lines, as it offers a clear way to showcase relationships expressed by equations.

linear equation graphing

Graphing linear equations involves drawing a straight line on the coordinate plane that satisfies a given equation, often in the form \(Ax + By = C\). This process uses intercepts or other known points to plot the linear function accurately.

To graph a linear equation like \(2y - x = 2\), follow these steps:

• Find the x-intercept by setting y to 0 and solving for x.
• Identify the y-intercept by setting x to 0 and solving for y.
• Plot these intercepts on the coordinate plane.
• Draw a straight line through the plotted intercepts.

This line represents all solutions to the equation. Graphing helps in visualizing the solutions and understanding the behavior of the function across the plane. Mastering linear equation graphing is a crucial skill for interpreting and analyzing mathematical relationships.