Problem 40 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=-3 x+6$$

Step-by-Step Solution

Answer

The graph is a straight line with y-intercept at (0, 6) and x-intercept at (2, 0).

Step 1: Understanding the Equation

The given equation is in the form of a linear equation: \( y = -3x + 6 \). This represents a straight line with a slope of -3 and a y-intercept of 6.

Step 2: Finding the Y-Intercept

To find the y-intercept, set \( x = 0\) and solve for \( y \): \( y = -3(0) + 6 = 6 \). Therefore, the y-intercept is \( (0, 6) \).

Step 3: Finding the X-Intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \): \( 0 = -3x + 6 \). Solving for \( x \) gives \( 3x = 6 \), so \( x = 2 \). Therefore, the x-intercept is \( (2, 0) \).

Step 4: Plotting the Intercepts

Mark the intercepts on the graph: the y-intercept at \( (0, 6) \) and the x-intercept at \( (2, 0) \).

Step 5: Drawing the Line

Draw a straight line through the points \( (0, 6) \) and \( (2, 0) \). This line represents the graph of the equation \( y = -3x + 6 \).

Step 6: Labeling the Intercepts

Label the intercepts on the graph as \( (0, 6) \) and \( (2, 0) \).

Key Concepts

y-intercept

The y-intercept is the point where the line crosses the y-axis. Simply put, it's where the value of x is zero. For the equation given, oindent \t \( y = -3x + 6 \),oindent setting \( x = 0 \) helps find this value. When you substitute \( x = 0 \) into the equation, you get:

x-intercept

The x-intercept is the point where the line crosses the x-axis. This is where the value of y is zero. For the equation given, oindent \t \( y = -3x + 6 \),oindent setting \( y = 0 \) helps find this value. Solving the equation with \( y = 0 \) provides the x-intercept at (2, 0).

slope

The slope of a line tells you how steep the line is. It represents the rate of change of y with respect to x. For the line represented by the equation \t \( y = -3x + 6 \),the slope is the coefficient of x, which is -3. A slope of -3 means that for every 1 unit increase in x, y decreases by 3 units. This negative slope means that the line goes downward from left to right.