Problem 8 - Algebra 1

Question

Find the slope and the y-intercept of the graph of the equation. $$ y=-1.5 x $$

Step-by-Step Solution

Answer

The slope of the graph is -1.5 and the y-intercept is 0.

Step 1: Identify the slope

The slope, \( m \), is the coefficient of \( x \) in the equation. Therefore, the slope of the equation is -1.5.

Step 2: Identify the y-intercept

The y-intercept, \( c \), is the constant in the equation. Since no constant is present, it defaults to 0. Therefore, the y-intercept of the equation is 0.

Key Concepts

Slope

The slope of a linear equation is a crucial concept in understanding how lines behave on a graph. It is denoted by the letter \( m \) and represents the steepness or incline of the line. The slope is also defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. If you are looking at the equation of a line like \( y = mx + c \), the number multiplying the \( x \) is the slope.In the equation \( y = -1.5x \), the slope is \(-1.5\). This negative value tells us that the line slopes downwards from left to right:

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.

The steeper the slope, the more vertical the line is. In the case of our equation, for every 1 unit increase in \( x \), the value of \( y \) decreases by 1.5 units. This helps in predicting the line's direction on a graph.

Y-intercept

The y-intercept is another fundamental part of linear equations. It indicates where the line crosses the y-axis on a graph. In the general equation \( y = mx + c \), the \( c \) represents the y-intercept. This is the point at which the line intersects the vertical axis, or where \( x = 0 \).In the equation \( y = -1.5x \), no constant term is included, meaning \( c = 0 \). Thus, the y-intercept is \( 0 \). This means the line passes through the origin, which is the point \((0, 0)\) on the graph:

• If \( c \) were positive, the line would intersect the y-axis above the origin.
• If \( c \) were negative, it would intersect below the origin.

Identifying the y-intercept is useful because it provides a fixed starting point to draw or interpret the line on a graph.

Graphing

Graphing linear equations is an excellent way to visually understand how equations behave. For the equation \( y = -1.5x \), graphing begins by noting the slope and y-intercept that we just discussed.1. Start at the y-intercept, which is \( 0 \) for this equation. Place a point on the origin, \( (0, 0) \).2. Use the slope to identify the next point. Since the slope is \(-1.5\), move down 1.5 units and 1 unit to the right from the origin. Place another point here.3. Draw a straight line through these points.With these steps:

• You can visualize how the line decreases as \( x \) values increase.
• Understanding this will help you see how different slopes and y-intercepts alter the line's path.

Graphing brings the math to life, making abstract numbers and formulas relatable and tangible.