Problem 43 - Elementary Algebra

Question

Determine the slope and \(y\) -intercept of the lines. $$ 3 y+3 x=1 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -1, and the y-intercept is 1/3.

Step 1: Rewrite the equation in slope-intercept form

First, we'll rewrite the given equation in the form y = mx + b. $$ 3y + 3x = 1 $$ To do this, we want to isolate y on the left side of the equation. Start by subtracting 3x from both sides of the equation to get: $$ 3y = -3x + 1 $$ Now, divide both sides of the equation by 3: $$ y = -x + \frac{1}{3} $$ We have successfully written the equation in slope-intercept form.

Step 2: Identify the slope

Now that the equation is in the form y = mx + b, we can simply read the slope from the equation. The slope, m, is the coefficient of x (the number in front of x). In our equation, y = -x + 1/3, we can see that the coefficient of x is -1. Therefore, the slope is: $$ m = -1 $$

Step 3: Identify the y-intercept

We can also read the y-intercept, b, directly from the equation in slope-intercept form. The y-intercept is the constant term (the number added or subtracted from x) in the equation. In our equation, y = -x + 1/3, the constant term is 1/3. Therefore, the y-intercept is: $$ b = \frac{1}{3} $$

Step 4: Present the solution

With the slope and y-intercept identified, we can present the solution: The slope of the line is -1, and the y-intercept is 1/3.

Key Concepts

Understanding Linear Equations

Linear equations, like the one given in the exercise, are equations of the first degree. This means that the highest power of any variable in the equation is one. These equations can generally be written in the form of \( ax + by = c \), where \( x \) and \( y \) are variables, and \( a \), \( b \), and \( c \) are constants. Linear equations often represent straight lines when graphed on a coordinate plane. These lines can have different slopes and y-intercepts, determining their inclination and where they intersect the y-axis, respectively. The primary goal in dealing with linear equations is often to rearrange them into a format that is easy to interpret, such as the slope-intercept form.

The Process of Isolating Variables

To solve linear equations or rearrange them into different forms, one must learn to isolate variables. This essentially means having one variable alone on one side of the equation. The standard form we aim for with linear equations involving \( y \) is \( y = mx + b \), known commonly as slope-intercept form. To isolate \( y \), we employ basic algebraic operations:

• Subtract or add terms to both sides to move constants or \( x \) terms
• Divide or multiply throughout to simplify coefficients

In our exercise, we subtracted \( 3x \) from both sides and then divided everything by 3 to isolate \( y \). This left us with \( y = -x + \frac{1}{3} \), a clear slope-intercept form.

Determining the Slope

The slope of a line is a measure of its steepness or how tilted it appears on the graph. When an equation is in the slope-intercept form \( y = mx + b \), the slope is represented by \( m \). It is crucial to understand what slope represents:

• A positive slope means the line ascends as \( x \) increases.
• A negative slope indicates the line descends as \( x \) grows.
• A slope of zero means the line is horizontal.

For the equation \( y = -x + \frac{1}{3} \), the coefficient of \( x \) is \(-1\). Thus, the slope \( m \) is \(-1\), indicating that as \( x \) increases, \( y \) decreases, giving the line a downward tilt.

Identifying the Y-Intercept

The y-intercept is a crucial element because it tells us where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the \( b \) represents the y-intercept. Finding the y-intercept is straightforward. It is the constant term in the equation, the number without an \( x \) attached. Practically, the y-intercept is the value of \( y \) when \( x \) is zero. For the equation \( y = -x + \frac{1}{3} \), the constant term is \( \frac{1}{3} \). Therefore, the y-intercept is \( \frac{1}{3} \). This means the line crosses the y-axis at the point \( (0, \frac{1}{3}) \), indicating where \( y \) meets the axis with \( x = 0 \).