Problem 38 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given point and has the given slope. $$ (3,0), m=-4 $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the point (3,0) and has a slope of -4 is \( y = -4x + 12 \).

Step 1: Insert given values.

Start by inserting the values for the point (x,y) = (3,0) and the slope m = -4 into the equation. This gives \( 0 = -4*3 + b \).

Step 2: Solve for the y-intercept b

After simplifying you get \( 0 = -12 + b \). Solve for b by adding 12 to both sides of the equation which results in \( b = 12 \)

Step 3: Write the final equation.

Insert the solved b into the slope-intercept form \( y = mx + b \). You get the final equation \( y = -4x + 12 \).

Key Concepts

Linear Equations

A linear equation is an equation that models a straight line when graphed on a coordinate plane. It describes a linear relationship between two variables, usually represented as "x" and "y". The general form for a linear equation is\( ax + by = c \), where "a", "b", and "c" are constants.

However, there is a more useful form for linear equations, especially when you want to easily identify the line's characteristics: the slope-intercept form. The slope-intercept form is \( y = mx + b \). In this form, "m" represents the slope of the line, and "b" indicates the y-intercept. This format makes it straightforward to graph the equation and understand the relationship it describes.

• "Linear" implies that the function forms a straight line.
• The names of the terms (slope and intercept) give insight into the line's behavior.
• This form simplifies finding solutions to related problems, such as determining the point of intersection between lines.

Understanding linear equations is a fundamental aspect of algebra, as they recur in various math problems and real-world applications.

Slope

The slope of a line is a measure representing how steep the line is. It's a vital part of linear equations, as it defines the rate of change between the "y" values and the "x" values. In a graph, the slope is calculated as the "rise over run" or the change in "y" over the change in "x". The formula for slope "m" is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

In the slope-intercept form \( y = mx + b \), "m" is the slope. You would typically substitute known values into this equation to figure out the line's behavior.

• A positive slope means the line rises as "x" increases.
• A negative slope indicates the line falls as "x" rises, as seen in the step-by-step solution with \( m = -4 \).
• A zero slope results in a horizontal line, suggesting no change in "y" as "x" changes.
• An undefined slope (division by zero) implies a vertical line.

Understanding the slope helps in predicting how the line behaves as it extends across the graph.

Y-Intercept

The y-intercept of a line is where the line crosses the y-axis. It's a point on the graph where "x" equals zero. In the slope-intercept equation \( y = mx + b \), the y-intercept is represented by "b". This value indicates where the line will intersect the y-axis, and it helps in graphing the line quickly.

In the step-by-step solution provided, by substituting the given slope and point into the slope-intercept form, the y-intercept was calculated as 12. This means the line intersects the y-axis at the point (0, 12).

• The y-intercept is crucial for determining where a line starts on the y-axis.
• It provides a reference point for drawing the remainder of the line using the slope.
• In practical terms, the y-intercept can represent an initial value or condition before changes start to occur as indicated by the slope.

Calculating the y-intercept gives a complete view of the linear equation, outlining where it begins and how it should look across the graph.