Problem 61 - 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. $$6 x-5 y=-30$$

Step-by-Step Solution

Answer

The points are (0, 6) and (5, 12). The slope is \(\frac{6}{5}\).

Step 1: Rewrite Equation

Start by rewriting the given equation in a form that's easier to analyze. We'll solve for \(y\) in terms of \(x\). Begin with the given equation: \[ 6x - 5y = -30 \] Rearrange the equation to solve for \(y\): \[ -5y = -6x - 30 \] Divide every term by \(-5\): \[ y = \frac{6}{5}x + 6 \]

Step 2: Find Two Points on the Line

Choose two values for \(x\), and solve for \(y\) to find two points on the line. Let's choose \(x = 0\) and \(x = 5\). - If \(x = 0\): \[ y = \frac{6}{5}(0) + 6 = 6 \] Point: \((0, 6)\)- If \(x = 5\): \[ y = \frac{6}{5}(5) + 6 = 6 + 6 = 12 \] Point: \((5, 12)\)

Step 3: Calculate the Slope

The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the coordinates of the points \((0, 6)\) and \((5, 12)\) into the formula: \[ m = \frac{12 - 6}{5 - 0} = \frac{6}{5} \] Thus, the slope of the line is \(\frac{6}{5}\).

Key Concepts

Finding Coordinates

When working with equations, especially linear ones, it's important to understand how to find the coordinates of points that lie on the line. Coordinates are pairs of numbers

• the first number representing the position on the x-axis
• the second on the y-axis

To determine these points on our line, we start by choosing random values for the x-coordinate. These can be any real numbers, but choosing simpler numbers like 0 or 1 often makes the work easier. These chosen x-values are plugged into the equation, and we solve for y. This gives us a complete coordinate, (x, y), for a point on the line.
For example, in our given equation, we choose:

• If \(x = 0\): solving the equation yields \(y = 6\), thus the point is \((0, 6)\).
• If \(x = 5\): solving the equation yields \(y = 12\), thus the point is \((5, 12)\).

These steps find you real points that you can even plot on a graph to visualize the line.

Equation of a Line

The equation of a line is like a recipe that tells us how to create a line graph on a plane. It describes the relationship between the x and y coordinates on a graph. Linear equations often appear in the format \(Ax + By = C\), but for graphing or understanding the line, we usually transform it to the slope-intercept form \(y = mx + b\). Here,

• \(m\) represents the slope of the line,
• \(b\) shows where the line crosses the y-axis (the y-intercept).

In our exercise, we had the equation \(6x - 5y = -30\). By rearranging this to find y in terms of x, we expressed it as \(y = \frac{6}{5}x + 6\).
This new form gives us all the insight we need to both understand the line's steepness and where it resides on the coordinate plane.

Graphing Linear Equations

Graphing linear equations involves plotting the line they represent on a coordinate plane. Once you have the slope-intercept form of the line, the task becomes straightforward.
First, identify two or more points on the line by either directly using the slope and y-intercept or calculating additional points by choosing values for x and solving for y. In our example, the point \((0, 6)\) is the y-intercept, which is a great starting point.
Use the slope, \(\frac{6}{5}\), to determine how the line increases as you move along the x-axis. This slope means that for every 5 units you move right on the x-axis, you move 6 units up on the y-axis. Start plotting from the y-intercept and mark the next point using this slope. The second point we used, \((5, 12)\), can be derived using this method.
Finally, connect these points with a straight line. This visual representation helps you better understand the relationships described by the equation.