Problem 64 - Elementary Algebra

Question

Find the coordinates of two points on the given line, and then use those coordinates to find the slope of the line. $$y=-2 x+5$$

Step-by-Step Solution

Answer

The slope of the line is \(-2\).

Step 1: Choose a Value for x

Let's find two points on the given line by choosing values for \(x\). Start by choosing \(x = 0\).

Step 2: Calculate y for x=0

Substitute \(x = 0\) into the equation \(y = -2x + 5\). This gives \(y = -2(0) + 5 = 5\). So, the point is \((0, 5)\).

Step 3: Choose Another Value for x

Now, choose another value for \(x\), for instance, \(x = 3\).

Step 4: Calculate y for x=3

Substitute \(x = 3\) into the equation \(y = -2x + 5\). This gives \(y = -2(3) + 5 = -6 + 5 = -1\). So, the second point is \((3, -1)\).

Step 5: Calculate the Slope using the Two Points

Use the two points \((0, 5)\) and \((3, -1)\) to find the slope \(m\). The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the values: \(m = \frac{-1 - 5}{3 - 0} = \frac{-6}{3} = -2\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a fascinating branch of mathematics that connects geometric figures with algebraic equations. This connection allows us to describe geometric shapes and properties using a coordinate plane. A coordinate plane is formed by two perpendicular lines, the x-axis and y-axis, which divide the plane into four quadrants. This setup makes it possible to precisely locate points using pairs of numbers, called coordinates.

• The first number in the pair is the x-coordinate, dictating how far the point is from the y-axis.
• The second number is the y-coordinate, dictating how far the point is from the x-axis.

Vegetably, any line or curve can be graphed on this coordinate plane by plotting points defined by equations in terms of x and y variables. Understanding this is crucial in solving problems involving lines, as it provides us a visual method to assess equations and their transformations. With this in mind, let's explore how this plays a role in constructing and interpreting linear equations.

Linear Equations

Linear equations are foundational in understanding algebra and serve as a simple way to describe lines in coordinate geometry. A linear equation in two variables, typically x and y, represents a straight line when graphed on a coordinate plane. The general form of a linear equation is given by:\[ y = mx + c \]where \(m\) is the slope of the line, and \(c\) is the y-intercept, which is the point where the line crosses the y-axis.

• The slope \(m\) informs us how steep the line is and the direction it slants.
• The y-intercept \(c\) specifies the exact height on the y-axis where the line makes its entry.

In our exercise, the equation \(y = -2x + 5\) is already in this linear form. Here, the slope \(-2\) tells us that for every unit increase in x, y decreases by 2 units. The y-intercept 5 indicates that the line passes through the point (0, 5) on the y-axis. This intuitive understanding simplifies finding the slope of a line given its equation and helps in plotting it correctly on a graph.

Slope of a Line

The slope of a line is a crucial element that defines its steepness and direction. To find the slope between two distinct points on a line, such as \((x_1, y_1)\) and \((x_2, y_2)\), we employ the slope formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula calculates the "rise" (the change in y-values) over the "run" (the change in x-values), giving us a ratio that reveals the tilt of the line.

• If the result is positive, the line rises as you move from left to right.
• If the result is negative, the line falls.
• A zero slope indicates a horizontal line, whereas an undefined slope indicates a vertical line.

When using this formula in our example with points (0, 5) and (3, -1), we substitute as follows:\[ m = \frac{-1 - 5}{3 - 0} = \frac{-6}{3} = -2 \]The negative slope of -2 means the line decreases two units in y for every one unit it moves to the right along the x-axis. Understanding the slope helps us interpret the movement and angle of a line much more clearly.