Problem 33 - Algebra 1

Question

Graph the points and draw a line through them. Write an equation in slope- intercept form of the line that passes through the points. $$ (-5,2),(6,1) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (-5,2) and (6,1) is \( y = -1/11x + 17/11\).

Step 1: Determine the Slope

To find the slope of the line that passes through the points (-5,2) and (6,1), we use the formula \(m = (y_2 - y_1) / (x_2 - x_1)\), where \(m\) is the slope, \((x_1, y_1)\) is the first point and \((x_2, y_2)\) is the second point. By substitution: \(m = (1 - 2) / (6 - (-5)) = -1 / 11\).

Step 2: Identify the Y-Intercept

The y-intercept b is the y-coordinate where the line crosses the y-axis. We can choose any one of the given points to find b, as the line passes through both. We'll use the point (6,1): 1 = (-1 / 11 * 6) + b. Solving for b, we get \( b = 1 + 6/11 = 17/11\).

Step 3: Write the Equation

Now that we know the slope m and the y-intercept b, we can write the equation of the line using the slope-intercept form: \(y = mx + b\). Substituting m = -1/11 and b = 17/11, our finished equation is \( y = -1/11x + 17/11\).

Step 4: Graphing the Line

To graph the line plot the y-intercept point (0,17/11) on the graph. The slope is -1/11, meaning that for every 1 unit increase in x, the y-value decreases by 1/11. Using this, draw the line that passes through the y-intercept and the other points (-5,2) and (6,1).

Key Concepts

Slope Calculation

Slope is an essential concept in algebra describing the steepness or incline of a line. It is often expressed as a ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. The formula to calculate it is:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \),
where \(m\) represents slope, and \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. This ratio gives a clear indication of the line's direction and steepness. In the example given, with the points (-5,2) and (6,1), the slope is calculated as
\(m = \frac{1 - 2}{6 - (-5)} = -\frac{1}{11}\).
Negative slope, such as this, indicates the line tilts downwards as it moves from left to right. Understanding how to calculate slope is crucial for graphing and interpreting linear functions.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. It is essential in graphing and understanding the starting value of the dependent variable when the independent variable is zero. To find the y-intercept, once we know the slope, we use one of the given points and the slope to solve for \(b\) in the equation \(y = mx + b\).
In the problem, using the point (6,1) with slope \(-\frac{1}{11}\), we set up the equation:
\(1 = -\frac{1}{11}(6) + b\),
and solve to find \(b = \frac{17}{11}\). This is where the line will cross the y-axis, and it becomes a pivotal point for graphing the entire line.

Graphing Lines

Graphing lines is a visual way to represent a linear equation and inspect the relationship between two variables. To graph a line, you usually need two things: a point and the slope. The first step is to plot the y-intercept, which is
\((0, \frac{17}{11})\)
in our example. Then, from the y-intercept, you apply the slope to find another point on the line.

Using the Slope

With the slope \(-\frac{1}{11}\), we move horizontally to the right 1 unit (because the run is 1), and vertically down \(\frac{1}{11}\) units (since the rise is negative, indicating a downward direction) from the y-intercept to mark the second point. Connect these two points with a straight edge, and the line is fully graphed. Graphing is a helpful way to visualize the function and confirm the accuracy of algebraic solutions.

Slope-Intercept Form

The slope-intercept form is a straightforward way to write linear equations and is represented by the formula:
\(y = mx + b\),
where \(m\) is the slope, and \(b\) is the y-intercept. This form is widely used because it clearly shows how much the y value changes with each increment of x (the slope) and at which point the line crosses the y-axis (the y-intercept). Once you have the slope and y-intercept, as in our exercise with
\(m = -\frac{1}{11}\) and \(b = \frac{17}{11}\),
plugging these values into the equation gives the desired function:
\(y = -\frac{1}{11}x + \frac{17}{11}\).
Such an equation pinpoints every single point on the line and provides the basis for graphing it accurately. The slope-intercept form is especially useful in real-world applications where the relationship between two quantities needs to be analyzed or predicted.