Problem 8 - Contemporary Precalculus

Question

Find a number t such that the line passing through the two given points has slope -2. $$(1, t) ;(-2,4)$$

Step-by-Step Solution

Answer

Question: Given two points (1, t) and (-2, 4) and the slope of the line that passes through them is -2, find the value of t. Answer: The value of t is -2.

Step 1: Write down the formula for the slope of a line

The formula for calculating the slope (m) of a line using coordinates of two points `\((x_1, y_1), (x_2, y_2)\)` is: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Step 2: Plug in the given values of points and slope

Now, let's plug in the given values into the formula. The points are \((1, t)\) and \((-2, 4)\), and the slope is -2. $$-2 = \frac{4 - t}{-2 - 1}$$

Step 3: Simplify the equation

Now, we'll simplify the equation: $$-2 = \frac{4 - t}{-3}$$

Step 4: Solve for t

To solve for t, we first multiply both sides with -3: \(-2 \times -3 = \frac{4 - t}{-3} \times -3\) Now, we get: $$6 = 4 - t$$ Next, subtract 4 from both sides to isolate t: $$6 - 4 = 4 - t - 4$$ Therefore, we find that: $$t = -2$$

Step 5: Write the final answer

So the number t is -2, and the point \((1, t)\) becomes \((1, -2)\). The line with a slope of -2 passes through the points \((1, -2)\) and \((-2, 4)\).

Key Concepts

Understanding Coordinates

Coordinates are fundamental in geometry. They define the position of a point on a plane using a pair of numbers: \(x\) and \(y\). These numbers are called the x-coordinate and y-coordinate, respectively. The x-coordinate tells us how far along the horizontal axis the point is, while the y-coordinate tells us how far up or down it is along the vertical axis.

For instance, the coordinates \((1, t)\) and \((-2, 4)\) mean:

• The point \(1, t\) is located 1 unit to the right of the origin (along the x-axis) and t units up (or down) the y-axis.
• The point \((-2, 4)\) is 2 units to the left of the origin and 4 units up the y-axis.

Understanding this helps us place points correctly in the coordinate plane and solve problems involving geometrical shapes like lines.

Linear Equations and the Slope of a Line

Linear equations are mathematical expressions that form straight lines when graphed on a coordinate plane. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

The slope of a line describes its steepness and direction:

• A positive slope climbs up from left to right.
• A negative slope descends from left to right.
• If the slope is zero, the line is horizontal.

In the context of our problem, knowing the slope is essential to find unknown point values such as t. Using the point-slope formula, \(m = \frac{y_2 - y_1}{x_2 - x_1}\), allows us to use given coordinates and the slope to set equations and solve for unknowns.

Algebraic Manipulation

Algebraic manipulation involves rearranging and solving equations to find unknown values. It is an essential skill in many math-related topics.

Here's how it was used in our problem:

• We started with the equation \(-2 = \frac{4 - t}{-3}\).
• To ensure \(t\) is alone, we multiplied both sides by -3, which clears out the fraction.
• This gave us the equation: \(6 = 4 - t\).
• Finally, to isolate \(t\), we subtracted 4 from both sides, resulting in: \(t = -2\).

Mastering these techniques is key to effectively solving equations, especially when dealing with linear equations and coordinate geometry.