Problem 47 Find the $x$ - and $y$ -inte... [FREE SOLUTION]

Chapter 2: Problem 47

Find the $x$ - and $y$ -intercepts and use them to graph the following functions. $$2 x+3 y=2$$

Short Answer

Expert verified

x-intercept: (1, 0), y-intercept: (0, 2/3).

Step by step solution

Finding the x-intercept

To find the x-intercept, set the y-value to zero in the equation, since the x-intercept is where the line crosses the x-axis. The equation becomes:\[ 2x + 3(0) = 2 \]Simplifying this, we get:\[ 2x = 2 \]Dividing both sides by 2, we find:\[ x = 1 \]Thus, the x-intercept is at the point $(1, 0)$.

Finding the y-intercept

To find the y-intercept, set the x-value to zero, since the y-intercept is where the line crosses the y-axis. Substitute x = 0 into the equation:\[ 2(0) + 3y = 2 \]This simplifies to:\[ 3y = 2 \]Dividing both sides by 3, we find:\[ y = \frac{2}{3} \]Thus, the y-intercept is at the point $(0, \frac{2}{3})$.

Plotting the intercepts on the graph

Now that we have found the intercepts $(1, 0)$ for x and $(0, \frac{2}{3})$ for y, we can plot these two points on the Cartesian plane. The point $(1, 0)$ is on the x-axis and $(0, \frac{2}{3})$ is slightly above the x-axis on the y-axis.

Drawing the graph

With the two intercepts plotted, draw a straight line through these points. This line represents the graph of the equation $2x + 3y = 2$. Since it's a straight line, ensure it extends infinitely in both directions, maintaining the constant slope determined by the two points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercept

The x-intercept of a line is where it crosses the x-axis of a graph. To find it, you set the value of y to zero in the equation. This is because on the x-axis, the y-coordinate is always zero. For example, in the equation $2x + 3y = 2$, setting y to zero gives the equation $2x = 2$. Solving for x, you divide both sides by 2 and find $x = 1$. This tells us that the x-intercept is at the point $(1, 0)$, meaning the line crosses the x-axis at x = 1.

The x-intercept is crucial for understanding where the line meets the horizontal axis.
When plotting a graph, it's one of the key points to determine the direction and slope of the line.

y-intercept

The y-intercept is the point where a line crosses the y-axis. To determine it, you set x to zero in the equation because any point on the y-axis has an x-coordinate of zero. Taking our example $2x + 3y = 2$, substitute zero for x to get $3y = 2$. Solving for y, divide each side by 3, resulting in $y = \frac{2}{3}$. Thus, the y-intercept is $(0, \frac{2}{3})$. This shows that the line intercepts the y-axis just a little below y = 1.

The y-intercept reveals where the line meets the vertical axis.
It helps provide a second key point needed to plot the graph accurately.

graphing linear equations

Graphing linear equations involves plotting points on a graph and drawing a line through them. For a linear equation like $2x + 3y = 2$, you determine intercepts first. Once the x-intercept $(1, 0)$ and y-intercept $(0, \frac{2}{3})$ are found, plot these on a graph.

Mark the x-intercept on the x-axis and the y-intercept on the y-axis.
Draw a line through these points. This straight line is the graphical representation of your equation.

Ensure the line is extended in both directions as it continues infinitely. The slope of the line remains consistent between any two points on a straight line.

Cartesian plane

A Cartesian plane is a two-dimensional plane for graphing mathematical functions, consisting of a horizontal x-axis and a vertical y-axis. Each point on the plane is described by a pair of numerical coordinates $(x, y)$. These coordinates demonstrate a point's distance from the axes.

The origin of the plane is $(0, 0)$, where the x-axis and y-axis intersect.
Finding intercepts places critical points on this plane, helping to graph equations accurately.

In graphing linear equations, the Cartesian plane provides the necessary structure to visualize how an equation corresponds to a straight line through plotted intercepts.

91影视

Find the \(x\) - and \(y\) -intercepts and use them to graph the following functions. $$2 x+3 y=2$$

Short Answer

Step by step solution

Finding the x-intercept

Finding the y-intercept

Plotting the intercepts on the graph

Drawing the graph

Key Concepts

x-intercept

y-intercept

graphing linear equations

Cartesian plane

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