Problem 27 - Contemporary Precalculus

Question

Find the slope and y-intercept of the line whose equation is given. $$3(x-2)+y=7-6(y+4)$$

Step-by-Step Solution

Answer

Answer: The slope of the line is $$\frac{3}{7}$$ and the y-intercept is $$-\frac{23}{7}$$.

Step 1: Simplify the given equation

Begin by simplifying the given equation. Distribute the 3 to (x-2) and -6 to (y+4): $$3x - 6 + y = 7 - 6y - 24$$

Step 2: Rewrite the equation in slope-intercept form

To rewrite the equation in slope-intercept form, let's first move all terms involving y to the left side and all other terms to the right side. First, add 6y to both sides of the equation: $$y + 6y = 3x - 6 + 7 - 24$$ Now, simplify both sides: $$7y = 3x - 23$$ Next, isolate y by dividing both sides of the equation by 7: $$y = \frac{3}{7}x - \frac{23}{7} $$ Now, the equation is in slope-intercept form (y = mx + b).

Step 3: Identify the slope (m) and y-intercept (b) from the equation

From the slope-intercept form of the equation, we can identify the slope (m) and y-intercept (b): Slope (m) = $$\frac{3}{7}$$ Y-intercept (b) = $$-\frac{23}{7}$$ Therefore, the slope of the line is $$\frac{3}{7}$$ and the y-intercept is $$-\frac{23}{7}$$.

Key Concepts

Linear Equations

Linear equations are equations that represent a straight line when plotted on a graph. They are usually in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. The term "linear" comes from the word "line," indicating that their graph is a straight line. These equations express a relationship with two variables, usually \( x \) and \( y \).

When working with linear equations, the main goal is often to solve for one variable in terms of the other. Today's exercise asks us to find specific characteristics of a line: the slope and the y-intercept. To do this, let's delve deeper into slope-intercept form, which simplifies identifying these characteristics.

Slope-Intercept Form

Slope-intercept form is a way of writing linear equations that makes it easy to identify the slope and y-intercept of the line. This form is written as \( y = mx + b \), where:

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

The slope \( m \) shows how steep the line is and the direction it slopes, either upwards or downwards. The y-intercept \( b \) provides the specific value of \( y \) when \( x = 0 \).

The main advantage of using slope-intercept form is its simplicity for graphing and understanding linear relationships. Rewriting equations into this form often involves isolating \( y \) on one side of the equation. Our exercise involves transforming the given equation into slope-intercept form by algebraically manipulating it to find the desired slope and y-intercept.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations in algebra. It includes operations such as distributing, combining like terms, and isolating variables to solve equations or rewrite them in different forms. In our example, algebraic manipulation was used to reformat the given equation into the slope-intercept form.

Let's see how these steps played out in our exercise:

• **Distributing:** The coefficients 3 and -6 were distributed to \((x-2)\) and \((y+4)\) respectively, which helped in breaking the expressions into separate terms.
• **Combining Like Terms:** Terms involving \( y \) were grouped together, which simplified the equation to provide easy manipulation.
• **Isolating Variables:** Finally, \( y \) was isolated on one side by dividing through by 7, putting the equation in the form \( y = \frac{3}{7}x - \frac{23}{7} \).

Through these steps, algebraic manipulation allows us to effectively determine the slope and y-intercept of the line from its equation.