Question

If the equation of a line is \(y=\frac{1}{2} x+3\), then mark on the graph the point where the line crosses the \(y\)-axis and the point where the line crosses the \(x\)-axis.

Short answer

The line crosses the y-axis at the point (0, 3) and crosses the x-axis at the point (-6, 0).

Step-by-step solution

Find the y-intercept

To find the y-intercept, set x=0 in the equation and solve for y: \(y = \frac{1}{2}(0) + 3\) \(y = 3\) So, the y-intercept is at the point (0, 3).

Find the x-intercept

To find the x-intercept, set y=0 in the equation and solve for x: \(0 = \frac{1}{2}x + 3\) \(-3 = \frac{1}{2}x\) Now, multiply both sides by 2 to isolate x: \(-6 = x\) So, the x-intercept is at the point (-6, 0).

Plot the points on the graph

Plot the points (0, 3) and (-6, 0) on a Cartesian plane (graph). The point (0, 3) is where the line crosses the y-axis, and the point (-6, 0) is where the line crosses the x-axis.

Key concepts

Understanding the X-Intercept

The x-intercept is a critical point where a line crosses the x-axis on a graph. For any linear equation like \[ y = mx + b \], the x-intercept can be found by setting \( y = 0 \) and solving for \( x \).
In the equation \( y = \frac{1}{2}x + 3 \), we find the x-intercept by substituting \( y = 0 \): \[0 = \frac{1}{2}x + 3\]
Solving this gives: \[-3 = \frac{1}{2}x\] Multiplying both sides by 2: \[-6 = x\].
The x-intercept is the point \((-6, 0)\) on the graph. This indicates where the line cuts through the x-axis. Identifying the x-intercept helps in understanding the behavior of the line as it provides key information about its position and slope.

• Set \( y = 0 \)
• Solve for \( x \)
• Mark point \((-6, 0)\)

Understanding the Y-Intercept

The y-intercept is the point on a graph where the line crosses the y-axis. It tells you where the line starts if you were to draw it from the intercept point. In the equation \( y = mx + b \), the parameter \( b \) represents the y-intercept. To find it, set \( x = 0 \) and solve for \( y \).
In our example, using \( y = \frac{1}{2}x + 3 \), substitute \( x = 0 \):\[y = \frac{1}{2}(0) + 3\]
This simplifies to \( y = 3 \), which means the y-intercept is at point \((0, 3)\).
This point gives you a place to start drawing the line, and it is essential for defining the line's slope and direction. Whenever tackling a new line equation, finding the y-intercept is always a good initial step.

• Set \( x = 0 \)
• Calculate \( y \)
• Mark point \((0, 3)\)

Navigating the Cartesian Plane

The Cartesian plane is a two-dimensional graph consisting of the horizontal x-axis and the vertical y-axis. It is used to plot points, lines, and curves, allowing us to visualize mathematical relationships. By understanding how to work with this plane, interpreting equations becomes more intuitive.
Here's a helpful breakdown:

• Axes: The horizontal line is called the x-axis, and the vertical line is the y-axis. They intersect at the origin \((0, 0)\).
• Quadrants: The plane is divided into four quadrants. Points in Quadrant I have positive x and y coordinates, Quadrant II has negative x and positive y, Quadrant III has both negative, and Quadrant IV has positive x and negative y.
• Plotting Points: Each point on the plane is identified by coordinates \((x, y)\). Moving right or left follows the x-axis, and moving up or down follows the y-axis.
• Lines: Linear equations can be graphed with points found using x- and y-intercepts, helping to visualize their slope and intersection on the Cartesian plane.

By using a Cartesian plane, we translate algebraic equations into geometric dimensions, making it easier to solve and analyze them visually.