Problem 62 - 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. $$7 x-6 y=-42$$

Step-by-Step Solution

Answer

The coordinates are (0, 7) and (6, 14); the slope is \(\frac{7}{6}\).

Step 1: Rearrange the equation to slope-intercept form

We start with the given equation of the line: \[7x - 6y = -42\] The first step is to rearrange this into the slope-intercept form, which is \(y = mx + b\). Start by isolating \(y\):\[-6y = -7x - 42\] Now, divide the entire equation by \(-6\) to solve for \(y\):\[y = \frac{7}{6}x + 7\] This equation shows that the slope \(m\) is \(\frac{7}{6}\) and the y-intercept is \(7\).

Step 2: Choose x-values to find corresponding y-values

To find the coordinates of two points on the line, substitute any two convenient values of \(x\) into the equation and solve for \(y\). Let's choose \(x = 0\) and \(x = 6\).

Step 3: Point 1 Calculation (x=0)

Substitute \(x = 0\) into the rearranged equation:\[y = \frac{7}{6}(0) + 7 = 7\]So the first point is \((0, 7)\).

Step 4: Point 2 Calculation (x=6)

Substitute \(x = 6\) into the rearranged equation:\[y = \frac{7}{6}(6) + 7 = 7 + 7 = 14\]So the second point is \((6, 14)\).

Step 5: Calculate the slope using the two points

The slope \(m\) of the line can also be calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the points \((x_1, y_1) = (0, 7)\) and \((x_2, y_2) = (6, 14)\) into the formula:\[m = \frac{14 - 7}{6 - 0} = \frac{7}{6}\]This confirms that the slope is \(\frac{7}{6}\).

Key Concepts

Coordinates of a Point

When you think of coordinates, picture a simple map with horizontal and vertical lines. Each point on this map is identified using a pair of numbers, called coordinates. These numbers are typically shown as

• The x-coordinate is the number that signifies a point's location on the horizontal (or x-axis).
• The y-coordinate indicates where the point lands on the vertical (or y-axis).

Together, ( x, y), these make up the complete coordinates of a point. In the problem we tackled, we found two points on a line by choosing different x-values and solving for the corresponding y-values. This helps in creating a clear picture of where the line passes on this map (or graph). By using coordinates like (0, 7) and (6, 14), we're essentially pinpointing exact spots the line crosses on our graph.

Linear Equation

A linear equation is one of the most straightforward expressions in math. It's called 'linear' because it represents a straight line when graphed on a coordinate system. The equation in our exercise, \[7x - 6y = -42\], needed some work to fit it into a more familiar form. This form is the "slope-intercept form," which is easier to interpret. A linear equation can include two variables, typically xy and y, and the sum of their terms.To understand linear equations:- **They help to define straight lines on a graph.**- **Each equation corresponds to one line.**- **Linear equations have solutions that are the points on the line.**By solving the equation, you can discover y-values for given x-values, helping us plot points on the graph accurately. Knowing how to manipulate and rearrange these equations is crucial in verifying the properties of the line, such as its slope and points.

Slope-Intercept Form

The slope-intercept form is a user-friendly way to express the equation of a straight line. Written as y = mx + b\, where:

• **m** is the slope of the line. It shows how steep the line is and which direction it goes.
• **b** is the y-intercept. It tells where the line crosses the y-axis.

Understanding this form makes graph reading and construction much more efficient.**Breaking it down:** The slope m, in\[y = \frac{7}{6}x + 7\], means the line rises 7 units for every 6 units moved horizontally to the right. The y-intercept b is 7, meaning this line will cross the y-axis at point (0, 7).Using the slope-intercept form, you can quickly identify the line's characteristics and even easily plot the line on a graph by starting at the y-intercept and following the slope.