91Ó°ÊÓ

The slope-intercept form of a linear equation is a very popular way to represent linear equations because of its simplicity and clarity. It is expressed as \( y = mx + b \). In this form, \( m \) stands for the slope of the line, while \( b \) is the y-intercept. These two key components make it very easy to sketch a line without needing to calculate many points.

• Slope - The slope is a measure of steepness and direction of the line. It tells us how much \( y \) (the vertical change) increases or decreases as \( x \) (the horizontal change) increases by 1 unit.
• Y-Intercept - The y-intercept is the point where the line crosses the y-axis. It shows the value of \( y \) when \( x \) equals zero.

Understanding this form helps in quickly graphing lines and finding their intersections. It's efficient and frequently used in solving various algebraic problems.

To find the y-intercept in a linear equation, we often make use of a given point and the slope. In the step-by-step solution to our problem, we had a point \((5, -3)\) and a slope \(m = \frac{3}{4}\). Using these, we can substitute them into the equation \( y = mx + b \) to solve for \( b \).

The solution for \( b \) involves simple plug-and-play:
1. Substitute the given \((x, y)\) values and the slope into the equation: \(-3 = \frac{3}{4}(5) + b\).2. Calculate \( \frac{3}{4} \times 5 \) to get \( \frac{15}{4} \).3. Rearrange to isolate \( b \) by subtracting \( \frac{15}{4} \) from both sides: \( b = -3 - \frac{15}{4} \).4. Simplify this to find \( b = -\frac{27}{4} \).Finding the y-intercept is a crucial step as it gives you a specific starting point on the graph from where you can apply the slope to find other points. This step typically involves basic arithmetic and handling fractions.

Once you have the equation of the line, plotting it becomes straightforward. The equation we derived, \( y = \frac{3}{4}x - \frac{27}{4} \), provides everything you need. Start with the y-intercept:
- Mark \( b = -\frac{27}{4} \) on the y-axis, which is approximately -6.75. This is your starting point on the graph.
From this point, use the slope to determine the next point. The slope \( \frac{3}{4} \) indicates:

• Rise 3 units up.
• Run 4 units to the right.

By following these steps, you plot the second point on the graph. Connect these points with a straight line to complete the graph. This visual representation of linear equations provides a clear way to see how variables relate, making it easier to digest complex algebraic concepts.