Problem 29 - Elementary Algebra

Question

Determine the slope and \(y\) -intercept of the lines. $$ y=4 x+10 $$

Step-by-Step Solution

Answer

Answer: The slope (m) of the line is 4, and the y-intercept (b) is 10.

Step 1: Identify the slope of the line

The slope (m) is the coefficient of the x-term in the slope-intercept form of the equation. In this case, the equation is: $$ y = 4x + 10 $$ The coefficient of the x-term is 4. Therefore, the slope of the line is 4.

Step 2: Identify the y-intercept of the line

The y-intercept (b) is the constant term in the slope-intercept form of the equation. In this case, the equation is: $$ y = 4x + 10 $$ The constant term is 10. Therefore, the y-intercept of the line is 10.

Step 3: Write the final answer

The slope of the line is 4, and the y-intercept is 10. The solution can be written as: - Slope (m) = 4 - Y-intercept (b) = 10

Key Concepts

Understanding Slope

The slope of a line is a measure of its steepness and direction. You can think of it as the 'rise over run,' which means how much the line goes up or down (rise) for how much it goes left or right (run). In the slope-intercept form of a linear equation, which is \( y = mx + b \), the slope is represented by the coefficient \( m \) of the \( x \) variable.

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• If the slope is zero, it is a horizontal line, indicating no rise or fall.

In our given equation \( y = 4x + 10 \), the number attached to \( x \) is 4, so the slope \( m \) = 4, suggesting that for each unit we move to the right, we move four units up. Understanding slope is crucial for interpreting how graphs behave and determining trends.

Exploring the Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. It is represented by \( b \) in the slope-intercept form, \( y = mx + b \). This point is always located at \( (0, b) \), meaning x is zero at the y-intercept.

• This is the starting point of the line when plotted on a graph.
• The value of \( b \) shows the position of the line vertically on the graph.
• If \( b \) is positive, as in \( y = 4x + 10 \), the line crosses the y-axis above the origin.

In our equation, \( b \) = 10, so the y-intercept is located at \( (0, 10) \). This means the line will cross directly at 10 units on the y-axis, setting the vertical starting point for the line.

Demystifying Linear Equations

Linear equations are mathematical statements expressing a straight line when graphed on a coordinate plane. The simplest and most direct form is the slope-intercept form, \( y = mx + b \), where you can easily read off the slope and y-intercept for graphing.

• The equation represents all the points that make up the line.
• With a known slope and y-intercept, you can quickly sketch the line even without further calculations.
• Linear equations like \( y = 4x + 10 \) model real-world phenomena, such as constant rates or trends.

Knowing how to interpret and use the slope and y-intercept helps understand changes and predict outcomes, which is invaluable in subjects from economics to physics. So, practicing the identification of slope and y-intercept strengthens understanding and application of linear equations.