Problem 52 - 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. $$7 x-3 y=21$$

Step-by-Step Solution

Answer

The points are (0, -7) and (3, 0), with a slope of \(\frac{7}{3}\).

Step 1: Convert Line Equation to Slope-Intercept Form

Start with the equation of the line: \(7x - 3y = 21\). To find points and the slope, first convert it to the slope-intercept form \(y = mx + b\). Solve for \(y\) by isolating it on one side.1. Subtract \(7x\) from both sides: \(-3y = -7x + 21\).2. Divide every term by \(-3\): \[ y = \frac{7}{3}x - 7 \] Now, the equation is in the form \(y = mx + b\), where \(m = \frac{7}{3}\) (the slope) and \(b = -7\) (the y-intercept).

Step 2: Choose Two Values for x

To find specific coordinates on the line, select two convenient x-values. Often, choosing values that simplify calculations is best, such as multiples of 3 because of the fraction in the slope. Let's choose \(x = 0\) and \(x = 3\).

Step 3: Calculate Corresponding y-values

Substitute the chosen x-values into the slope-intercept equation to find their corresponding y-values.1. For \(x = 0\): \[ y = \frac{7}{3}(0) - 7 = -7 \] Thus, the first point is \((0, -7)\).2. For \(x = 3\): \[ y = \frac{7}{3}(3) - 7 = 7 - 7 = 0 \] Thus, the second point is \((3, 0)\).

Step 4: Use Two Points to Calculate the Slope

The two points identified on the line are \((0, -7)\) and \((3, 0)\). To find the slope \(m\) using these points, apply the slope formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the coordinates:\[ m = \frac{0 - (-7)}{3 - 0} = \frac{7}{3} \]This confirms that the slope of the line is \(\frac{7}{3}\), which matches our converted slope from Step 1.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is incredibly useful in analyzing and graphing lines. It is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) denotes the y-intercept. This form provides a straightforward way to visualize how changes in \(x\) affect \(y\).

To convert any given equation to the slope-intercept form, you need to solve for \(y\), ensuring it stands alone on one side of the equation. For example, with an equation like \(7x - 3y = 21\), you would isolate \(y\) by initially subtracting \(7x\) from both sides and then dividing each term by \(-3\).

• This process will highlight the slope \(m = \frac{7}{3}\) and the y-intercept \(b = -7\).

This form is a foundational tool in understanding straight lines in coordinate geometry.

Linear Equations

Linear equations denote relationships where the highest power of the variable is one, illustrating a straight line when plotted on a graph. They have the general form of \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. In coordinate geometry, these equations make understanding and predicting linear relationships easy.

When working with linear equations, the goal often involves finding specific points that lie on the line. By choosing strategic values for \(x\), we can solve for \(y\) and locate these points. This helps in drawing the line or in cross-verifying data if given coordinates are part of it.

• For example, setting \(x = 0\) in \(7x - 3y = 21\) gives us the y-coordinate of the point where the line crosses the y-axis.
• Similarly, picking another value like \(x = 3\) helps find the corresponding \(y\) and provides clarity about the line's direction and incline.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, utilizes coordinates to define and describe geometric figures. This branch of mathematics enables us to translate algebraic equations, such as linear equations, into geometric interpretations through graphing.

In this context, the foundation of coordinate geometry lies in identifying and plotting points, lines, and slopes on a Cartesian plane. Each point is an \((x, y)\) coordinate derived from solving linear equations.

• Finding the slope of a line through two points, for example, can help determine the angle and direction of the line.
• Utilizing the formula \(\frac{y_2 - y_1}{x_2 - x_1}\) concretely displays how different points relate in terms of direction and steepness.

The application of coordinate geometry extends beyond simply identifying lines and points but becomes a critical tool in solving real-world geometric problems.