Problem 39 A student found the slope of the... [FREE SOLUTION] (2024)

Short Answer

slope formula

Understanding the slope formula is key to solving problems involving the steepness or incline of a line. The slope, often represented by the letter \(m\), measures how much a line rises or falls as it moves from one point to another. The general formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
. Here, \( y_2 - y_1 \) represents the vertical change (rise) between the two points, while \( x_2 - x_1 \) represents the horizontal change (run). By plugging the coordinates of the given points into this formula, you can determine the slope of the line that connects them.

point coordinates

In order to calculate the slope using the slope formula, you first need the coordinates of two points. Point coordinates are written as ordered pairs \((x, y)\), and they tell you the location of a point on a graph.
For example, in the given exercise, the points are \((-4,5)\) and \((2,7)\). '(-4,5)' means that moving 4 units to the left on the x-axis and then 5 units up on the y-axis will get you to the point, and '(2,7)' means moving 2 units to the right on the x-axis and 7 units up on the y-axis.
Plugging these coordinates into the slope formula lets us understand how steep the line joining these points is.

line equations

Once you have the slope, it's often used to write the equation of a line. The most common form is the slope-intercept form, written as:
\( y = mx + b \)
where \(m\) is the slope and \(b\) is the y-intercept (the point where the line crosses the y-axis).
Knowing the slope can help you describe the relationship between the x and y coordinates for every point on the line.
If you know one point on the line and the slope, you can also use the point-slope form of the line equation:
\( y - y_1 = m(x - x_1) \),
where \((x_1, y_1)\) is a known point on the line. This makes it easier to find the equation of a line and understand how it behaves graphically.

algebraic simplification

Algebraic simplification is crucial when calculating the slope and forming line equations. Simplifying expressions ensures calculations are easier and more accurate.
For example, in Step 3 of the solution, simplifying the numerator \( 7 - 5 \) gives \( 2 \). In Step 4, simplifying the denominator \( 2 - (-4) \) involves understanding that subtracting a negative is the same as adding: \( 2 + 4 \). This simplification results in \( 6 \).
Finally, dividing the numerator by the denominator in Step 5, \( \frac{2}{6} \) simplifies to \( \frac{1}{3} \).
Mastering these simplification steps avoids errors and makes the problem-solving process clear and straightforward.