Problem 46 - Understanding Elementary Algebra with Geometry

Question

Determine the slope of the line from its equation. $$5 x-6 y-7=0$$

Step-by-Step Solution

Answer

The slope is \( \frac{5}{6} \).

Step 1 - Rewrite the equation in slope-intercept form

To find the slope, rearrange the given equation into the slope-intercept form, which is \text{slope-intercept form} \text{is} y = mx + b. The given equation is :$$ 5x - 6y - 7 = 0 $$.

Step 2 - Isolate the \(y\)-term

Rewrite the equation to isolate the y-term on one side. Starting with: $$ 5x - 6y - 7 = 0,$$ Add 7 to both sides to get: $$ 5x - 6y = 7.$$

Step 3 - Solve for \(y\)

Now, get y by itself. Subtract 5x from both sides: $$ -6y = -5x + 7.$$ Then divide everything by -6: $$ y = \frac{5}{6}x - \frac{7}{6}.$$ The equation is now in the form $$ y = mx + b$$.

Step 4 - Identify the slope

The slope-intercept form is $$ y = mx + b$$ , where m is the slope. From the equation $$ y = \frac{5}{6}x - \frac{7}{6}$$, identify the slope m as \( \frac{5}{6} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a fundamental concept in algebra. It is expressed as \(y = mx + b\). This form allows you to easily identify the slope \(m\) and the y-intercept \(b\) of a line.

In the given exercise, our task is to convert the equation \(5x - 6y - 7 = 0\) into this form. The slope-intercept form makes it straightforward to determine the slope by turning the equation into a format where the coefficient of \(x\) directly represents the slope.

To do this:

• First, isolate the \(y\)-term on one side of the equation.
• Then, rearrange the remaining terms so the equation reads \(y = mx + b\).

In our case, once the equation is rearranged to \(y = \frac{5}{6}x - \frac{7}{6}\), we find the slope \(m\) is \frac{5}{6}\.

Solving Linear Equations

Solving linear equations is a crucial skill in algebra. It involves finding the value of the variable that makes the equation true.

In the step-by-step solution, solving the linear equation \(5x - 6y - 7 = 0\) involves several steps:

• Add or subtract terms to both sides of the equation move all terms involving \(x\) and constants to one side, and isolate the \(y\)-term on one side. For our example, we add 7 to both sides getting \(5x - 6y = 7\).


• Then, solve for the \(y\)-term by moving \(x\)-terms to the other side and dividing through by the coefficient of \(y\). In this case, divide by -6 to get \(y = \frac{5}{6}x - \frac{7}{6} \).

Using these steps systematically helps you solve the equation and rewrite it in slope-intercept form.

Rearranging Equations

Rearranging equations is essential to simplifying and solving them. This often includes moving terms from one side to the other, factoring, combining like terms, and isolating variables. In algebra, these manipulations help change the form of an equation for easier handling.

In our equation \(5x - 6y - 7 = 0\), we needed to isolate the \(y\)-term to rewrite the equation in slope-intercept form:

• First, move any constants to the other side of the equation \(5x - 6y = 7\)
• Then, rearrange to isolate \(y\) on one side \(-6y = -5x + 7\).
• Finally, divide every term by the coefficient of \(y\) (in this case, -6) to complete the rearrangement \(y = \frac{5}{6}x - \frac{7}{6}\).

Rearranging the equation this way prepares it for easy identification of the slope \ \frac{5}{6} \ and enhances our understanding of the line's properties.