Problem 33 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ 4 y=16 x+20 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 4 and the y-intercept is 5.

Step 1: Rewrite the given equation in slope-intercept form

First, we need to rewrite the given equation in the form of \(y=mx+b\). We have: $$ 4y = 16x + 20 $$ Divide both sides by 4: $$ y = 4x + 5 $$

Step 2: Identify the slope and y-intercept

Now that we have the equation in slope-intercept form, we can easily identify the slope and y-intercept. The slope (\(m\)) is the coefficient of x, which is 4. The y-intercept (\(b\)) is the constant term, which is 5. So, the slope of the line is 4 and the y-intercept is 5.

Key Concepts

Linear Equations

Linear equations are foundational in mathematics, representing relationships where variables change in a proportional manner. They have the general form of \( y = mx + b \), known as the slope-intercept form. Here, \( y \) and \( x \) are variables, while \( m \) and \( b \) are constants. The equation describes a straight line when graphed on a Cartesian plane.

Linear equations can be used to solve real-world problems, such as calculating the amount owed in a taxi fare, where the cost is proportional to the distance. In an equation, each term either has a constant multiplier or is a standalone constant, like \( 16x \) or \( +20 \) in the original equation. Simplifying linear equations often involves rearranging to isolate \( y \), which tells us the form of the equation and how the line behaves on a graph.

Slope

The slope of a line in a linear equation indicates how steep the line is. In the equation \( y = mx + b \), the slope is represented by \( m \).[Slope] gives the rate of change of \( y \) with respect to \( x \).

A positive slope means the line rises from left to right, while a negative slope means it falls. A slope of zero indicates a horizontal line, showing that \( y \) does not change as \( x \) changes. The steeper the line, the greater the magnitude of the slope. For example, in the equation \( y = 4x + 5 \), the slope is 4. This means for every unit increase in \( x \), \( y \) increases by 4 units. The slope is crucial in understanding the direction and angle of the line on a graph.

Y-Intercept

In the slope-intercept form \( y = mx + b \), the \( y \)-intercept is denoted by \( b \). This is the point where the line crosses the y-axis. At this point, \( x \) is zero.[Y-Intercept] is essentially the starting point of the line on the \( y \)-axis.

The \( y \)-intercept provides valuable insight into the initial condition of the line. For example, in the equation \( y = 4x + 5 \), the \( y \)-intercept is 5. This means that when \( x = 0 \), \( y \) is 5. This point (0, 5) is where the line intersects the y-axis on a graph. Understanding the \( y \)-intercept is helpful in applications like budgeting or planning, where it might represent an initial cost or fixed starting value.