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The y-intercept of a linear equation is where the graph of the equation crosses the y-axis. To find the y-intercept of the equation given, which is in the form of: \[y = -\frac{1}{4}x + 3\], set \(x = 0\) and solve for \(y\). In this equation, when \(x = 0\), we have: \[y = -\frac{1}{4}(0) + 3 = 3\].

Thus, the y-intercept is the point where \(x\) is 0 and \(y\) is 3. This can be written as the point (0, 3). It's crucial because it gives a starting point for drawing the graph of the equation on a coordinate plane.

For any linear equation in the form of \(y = mx + b\), the y-intercept is simply \(b\).

A linear graph represents a linear equation on a coordinate plane. The equation provided, \(y = -\frac{1}{4}x + 3\), forms a straight line when plotted.

To draw the linear graph, you need at least two points. However, having three or more points can ensure accuracy. In our example, the points (0, 3), (4, 2), and (-4, 4) were calculated and plotted. With these points, you can draw a straight line through them, as all the points from a linear equation lay on a straight line.

Here's a step-by-step method to graph a linear equation:

• Step 1: Plot the y-intercept on the graph.
• Step 2: Use the slope to determine another point on the line. For example, from the y-intercept (0, 3), move down 1 unit and right 4 units to find a new point at (4, 2).
• Step 3: Repeat to find a third point for accuracy. Move up 1 unit and left 4 units from the y-intercept to find (-4, 4).
• Step 4: Draw a straight line through these points.

This line represents all the solutions to the equation.

The slope of a line indicates its steepness and the direction in which it tilts. The slope is represented by \(m\) in the linear equation \(y = mx + b\). For the equation \(y = -\frac{1}{4}x + 3\), the slope \(m\) is \(-\frac{1}{4}\).

This slope means for every 4 units you move to the right along the x-axis, you move 1 unit down along the y-axis. This can be written as a ratio: rise over run. In our example:

• Rise: -1 (indicating a downward movement)
• Run: 4 (indicating movement to the right)

The negative slope indicates that the line falls as it moves from left to right. Understanding the slope is vital for quickly sketching the line and understanding how different linear equations behave.

By combining the y-intercept and the slope, you can accurately graph any linear equation. Place the y-intercept on the graph, and then use the slope to find other points to draw your line.