Problem 24 Find the slope and \(y\) -interc... [FREE SOLUTION] (2024)

Expert verified

In the context of linear equations, the slope is a measure of how steep a line is. It tells us how much the y-value (vertical) changes for a one-unit change in the x-value (horizontal).
The slope is often represented by the letter 'm'. In the slope-intercept form of a linear equation, which is written as \( y = mx + b \), 'm' is the slope.

To find the slope, we look at the coefficient of the x term. Slope can also be calculated by the rise-over-run formula, which is \[ m = \frac{\Delta y}{\Delta x} \]
This formula means the change in y divided by the change in x. In our example, after converting the equation to slope-intercept form \( y = x + 2 \). Here, the slope \( m = 1 \).

Key takeaways about slope:

• It measures the steepness of the line.
• Calculated as the ratio of the change in y to the change in x.
• A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.

The y-intercept is the point where the line crosses the y-axis. In other words, it's the value of y when x is zero. It's represented by 'b' in the slope-intercept form of the equation \( y = mx + b \).
To find the y-intercept, we simply look at the constant term in the equation. In our example, the equation \( y = x + 2 \) shows a y-intercept of 2. This means that when x is 0, y is 2.

Understanding the y-intercept can be very helpful:

• It shows where the line starts on the y-axis.
• Helps in graphing the line quickly.
• Gives a point of reference for the line.

Remember, the y-intercept is always found by setting x to 0 in the equation and solving for y.

Linear equations are mathematical statements that create a straight line when plotted on a graph. They usually come in the form \( ax + by = c \) or can be rewritten as \( y = mx + b \) (slope-intercept form).
The main feature of linear equations is that the relationship between the variables x and y is constant, meaning the graph will always be a straight line.

For example, in the equation \( 3x - 3y + 6 = 0 \), we can isolate y to get the slope-intercept form \( y = x + 2 \). This transformation makes it easier to understand the graph of the line.

Key points to remember about linear equations:

• They graph as straight lines.
• The slope-intercept form \( y = mx + b \) is very useful for understanding the slope and y-intercept.
• Linear equations represent linear relationships between variables.

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is used to express and solve equations and inequalities.
In our example, algebra helps us transform the given equation \( 3x - 3y + 6 = 0 \) into the slope-intercept form \( y = x + 2 \).

Understanding some basic concepts of algebra is essential:

• Variables: Symbols like x and y that represent numbers.
• Equations: Mathematical statements that show the equality of two expressions.
• Operations: Basic arithmetic (addition, subtraction, multiplication, division), but applied to variables and constants.

With algebra, we can manipulate equations and inequalities to find solutions to various problems. It's a fundamental tool for advanced mathematics and real-world problem-solving.