Problem 26 - Precalculus

Question

Find an equation for the line that is described, and sketch the graph. Write the final answer in the form \(y=m x+b ;\) (a) Passes through (-7,-2) and (0,0) (b) Passes through (6,-3) and has \(y\) -intercept 8 (c) Passes through (0,-1) and has the same slope as the line \(3 x+4 y=12\) (d) Passes through (6,2) and has the same \(x\) -intercept as the line \(-2 x+y=1\) (e) Has \(x\) -intercept -6 and \(y\) -intercept \(\sqrt{2}\)

Step-by-Step Solution

Answer

(a) \(y = \frac{2}{7}x\); (b) \(y = -\frac{11}{6}x + 8\); (c) \(y = -\frac{3}{4}x - 1\); (d) Info pending; (e) \(\sqrt{2}x + 6y = 6\sqrt{2}\).

Step 1 (a): Identify Points and Formulate Slope

We are given two points: \((-7, -2)\) and \((0, 0)\). The slope of the line is calculated using the formula for slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points: \(m = \frac{0 - (-2)}{0 - (-7)} = \frac{2}{7}\).

Step 2 (a): Use Point-Slope Form to Derive the Equation

With the slope \(m = \frac{2}{7}\) and a point \((0, 0)\) (which is also the origin), we use the point-slope form: \(y - y_1 = m(x - x_1)\). Substituting \((x_1, y_1) = (0, 0)\), we have \(y - 0 = \frac{2}{7}(x - 0)\), which simplifies to \(y = \frac{2}{7}x\).

Step 3 (b): Gather Information for Slope and Intercept

The point \((6, -3)\) and \(y\)-intercept 8 are given. The equation of the line will be of the form \(y = mx + 8\). To find \(m\), use the point \((6, -3)\) because it satisfies the equation: \(-3 = m(6) + 8\).

Step 4 (b): Solve for the Slope and Write the Equation

Solving the equation \(-3 = 6m + 8\) gives us: \(6m = -3 - 8 = -11\), thus \(m = -\frac{11}{6}\). Substituting \(m\) back, the equation is \(y = -\frac{11}{6}x + 8\).

Step 5 (c): Find the Slope of the Given Equation

The given line equation is \(3x + 4y = 12\). We need to find the slope \(m\). Rearranging gives \(4y = -3x + 12\), or \(y = -\frac{3}{4}x + 3\). Thus, the slope \(m = -\frac{3}{4}\).

Step 6 (c): Write the Equation Using Point and Slope

The new line passes through \((0, -1)\) and has slope \(-\frac{3}{4}\). Thus, using the equation format \(y = mx + b\) with intercept \(-1\), we find \(b = -1\). Therefore, the equation is \(y = -\frac{3}{4}x - 1\).

Step 7 (d): Find the x-intercept of the Given Line

The given line is \(-2x + y = 1\). To find the x-intercept, set \(y=0\): \(-2x = 1\), solving gives \(x = -\frac{1}{2}\).

Step 8 (d): Use the Same x-intercept for New Equation

The line must pass through \((6, 2)\) and intercept the x-axis at \(-\frac{1}{2}\). Use point-slope form with intercept: \(y - 2 = m(x - 6)\). Solve \((0,0)\): remainder to find the slope and write the equation.

Step 9 (e): Use x- and y-intercepts to Write the Equation

Given x-intercept \(-6\) and y-intercept \(\sqrt{2}\), the equation follows the intercept form: \(\frac{x}{-6} + \frac{y}{\sqrt{2}} = 1\). Multiplying throughout to clear fractions gives us: \(\sqrt{2}x + 6y = 6\sqrt{2}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line equation is one of the most frequently used formats when dealing with linear equations. It takes the form: \(y = mx + b\). Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept, which is where the line crosses the y-axis.
This form is extremely useful because it provides instant information about the line's behavior:

• \(m\) tells us whether the line ascends or descends as it moves from left to right. A positive \(m\) means the line goes up, while a negative \(m\) indicates it goes down.
• The y-intercept \(b\) is the point \((0,b)\), signifying where the line crosses the y-axis.

Let's look at an example: consider a line with equation \(y = 2x + 3\). Here:

• Slope \(m = 2\) means for every unit increase in \(x\), \(y\) increases by 2.
• Y-intercept \(b = 3\) so the line crosses the y-axis at \((0,3)\).

This easy-to-apply form makes plotting lines and understanding their directions a breeze.

Point-Slope Form

The point-slope form is perfect for writing an equation of a line when you know one point on the line and the slope. It is expressed as \(y - y_1 = m(x - x_1)\), where:

• \((x_1, y_1)\) is a point on the line
• \(m\) is the slope

This format is straightforward when you have a specific point and a known slope. Let's dive into how this works with an example. Suppose we have a point \((4, 5)\) and a slope \(m = \frac{1}{2}\):

• Substitute these values into the formula: \(y - 5 = \frac{1}{2}(x - 4)\)
• Distribute and simplify to convert to slope-intercept form: \(y = \frac{1}{2}x + 3\)

Thus, it's clear how this single starting point and a direction dictated by the slope can define an entire line. This form is often converted to the slope-intercept form for ease of graphing and further calculations.

Intercepts in Equations

Intercepts are where a line crosses the x-axis and y-axis, providing useful information about a line's position and orientation. An equation's intercepts are easily identified or calculated to convey critical data:

• The \(x\)-intercept is found by setting \(y = 0\) in the equation. For example, in \(3x + 4y = 12\), set \(y = 0\) to find \(x = 4\), meaning the x-intercept is \((4,0)\).
• The \(y\)-intercept involves setting \(x = 0\). Using the same equation, we find it by setting \(x = 0\), which results in \(y = 3\), hence the y-intercept is \((0,3)\).

These intercepts are essential for graphing as they provide quick reference points. Additionally, intercept-form equations like \(\frac{x}{a} + \frac{y}{b} = 1\) directly use intercepts \((a,0)\) and \((0,b)\) to plot lines efficiently and clearly. Remember, knowing intercepts simplifies the task of sketching or analyzing a linear equation, revealing where lines slice through each axis.