Problem 40 - Elementary Algebra

Question

Determine the slope and \(y\) -intercept of the lines. $$ 7 y+3 x=10 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -3/7 and the y-intercept is 10/7.

Step 1: Rewrite the equation in slope-intercept form (y = mx + b)

To do this, we will isolate y on one side of the equation: $$ 7y + 3x = 10 $$ Subtract 3x from both sides: $$ 7y = -3x + 10 $$ Divide by 7: $$ y = -\frac{3}{7}x + \frac{10}{7} $$

Step 2: Identify the slope and y-intercept

The equation is now in the form y = mx + b. We can see that: $$ m = -\frac{3}{7} $$ and $$ b = \frac{10}{7} $$ So, the slope of the line is -3/7 and the y-intercept is 10/7.

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra and describe a straight line on a coordinate plane. They are typically of the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. This form is known as the standard form.
Linear equations may also be represented in another popular format known as the slope-intercept form, \( y = mx + b \). This form explicitly shows the slope \( m \) and the \( y \)-intercept \( b \) of the line.
The process of converting from standard form to slope-intercept form involves isolating the \( y \) variable on one side of the equation. This transformation is particularly useful because it clearly provides vital information about the line's characteristics – the slope and the \( y \)-intercept.

• Standard form: \( ax + by = c \)
• Slope-intercept form: \( y = mx + b \)

Defining the Slope

The slope of a line is a measure of its steepness and is represented by the letter \( m \) in the slope-intercept form \( y = mx + b \). The slope indicates how much \( y \) increases or decreases as \( x \) increases by 1 unit.

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A zero slope means the line is horizontal and does not rise or fall.
• An undefined slope indicates a vertical line.

Using the conversion process from the original equation \( 7y + 3x = 10 \), we find that the slope \( m = -\frac{3}{7} \). This tells us that for every 7 units the line moves horizontally, it decreases by 3 units vertically. Therefore, the line is heading downwards.

Exploring the Y-Intercept

The \( y \)-intercept of a line gives the point where the line crosses the \( y \)-axis. This is represented by \( b \) in \( y = mx + b \). It provides crucial information, serving as an initial value of \( y \) when \( x \) is zero.
Finding the \( y \)-intercept involves setting \( x = 0 \) in the equation and solving for \( y \). In the equation \( y = -\frac{3}{7}x + \frac{10}{7} \), the \( y \)-intercept is \( \frac{10}{7} \). This means the line crosses the \( y \)-axis at \( y = \frac{10}{7} \).

• The \( y \)-intercept is the point \((0, b)\).
• It reveals where the line will start on the graph when plotting from the \( y \)-axis.

Understanding the \( y \)-intercept allows easier graphing of the line and helps predict its path through coordinate space.