Problem 46 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(-4 ;\) the \(y\) -intercept is 7

Step-by-Step Solution

Answer

The equation of the line in the slope-intercept form is \( y = -4x + 7 \)

Step 1: Understand the Slope-Intercept Form

First, understand the slope-intercept form of the equation of a line, which is \( y = mx + b \). Here, 'm' stands for the slope of the line and 'b' is the Y-intercept, the point where the line crosses the Y axis.

Step 2: Substitute the Given Values

Substitute the given values of the slope (-4) and y-intercept (7) into the slope-intercept equation. This leads to the equation \( y = -4x + 7 \).

Step 3: Write the Final Equation

The final equation of the line, by substituting the given slope and y-intercept, is \( y = -4x + 7 \).

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and represent straight lines when graphically plotted. A linear equation can be written in the form of \( y = mx + b \), which is known as the slope-intercept form. This form is particularly useful as it provides clear information about the line's slope and y-intercept.

Linear equations show a direct relationship between two variables, which means as one variable increases or decreases, the other does as well. Key characteristics:

• The degree of a linear equation is always one.
• Linear equations can be rearranged into different forms, like the standard form \( Ax + By = C \).
• The graph of a linear equation is always a straight line.

Understanding linear equations is crucial for solving many problems in algebra and calculus. They provide a foundation for understanding more complex mathematical concepts.

Slope

The slope is a measure of the steepness or incline of a line. In the slope-intercept form \( y = mx + b \), \( m \) represents the slope. It is a crucial part of a linear equation because it indicates how much \( y \) rises or falls as \( x \) increases by one unit.

Knowing the slope of a line helps determine its direction:

• If the slope is positive, the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A slope of zero indicates a horizontal line, showing no change in \( y \) with a change in \( x \).

The slope can be calculated using two points on the line: \((x_1, y_1)\) and \((x_2, y_2)\) with the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Understanding slopes helps in analyzing linear trends and patterns in data.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) is the y-intercept. It depicts the value of \( y \) when \( x \) equals zero. This point is crucial for quick sketching of graphs and understanding initial values in a context.

To identify the y-intercept from an equation:

• Locate the constant term \( b \) in the equation.
• This constant shows where the line will intersect the y-axis.

For example, if we have the equation \( y = -4x + 7 \):

• The y-intercept is \( 7 \), indicating the line crosses the y-axis at the point \((0, 7)\).

The y-intercept tells us the initial point or starting value of data represented by the equation and is invaluable in analyzing how changes occur from a starting point.