Problem 19 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ m=-4, y \text { -intercept }(0,0) $$

Step-by-Step Solution

Answer

Answer: The equation of the line in slope-intercept form is y = -4x.

Step 1: Write down the slope and y-intercept

We are given the slope m = -4 and y-intercept (0,0).

Step 2: Plug in the given slope and y-intercept in the slope-intercept form equation

We will now plug in m = -4 and the y-intercept coordinate (0,0) in the slope-intercept form equation: $$ y = mx + b $$

Step 3: Solve for b using the y-intercept

Since the y-intercept is (0,0), we can plug in these coordinates (x = 0, y = 0) into the equation and solve for b: $$ 0 = (-4)(0) + b $$ This simplifies to: $$ b = 0 $$

Step 4: Write the final equation of the line using the slope and y-intercept

Now that we have found the value of b, we can write down the final equation of the line: $$ y = -4x + 0 $$ or simply: $$ y = -4x $$ This is the equation of the given line in slope-intercept form.

Key Concepts

Equation of a Line

The equation of a line is a mathematical expression that describes all the points that lie on that line. In simpler terms, it tells us how the line behaves on a graph. The most common form to express this is the slope-intercept form, given by the equation: \[ y = mx + b \]This equation consists of two main parts:

• The slope \(m\), which indicates how steep the line is, and
• The y-intercept \(b\), which is where the line crosses the y-axis.

Knowing these two features allows you to graph the line easily. You can determine how a line will look just by examining the equation, making it a powerful tool in understanding linear relationships in mathematics.

Linear Equations

Linear equations are equations of the first degree, meaning they have no exponents higher than one. Their graphs are always straight lines. In the slope-intercept form, \[ y = mx + b \]the equation represents a linear relationship between the variables \(x\) and \(y\). Linear equations have characteristics that make them vital in different applications:

• They help in solving real-world problems by modeling everyday situations.
• They form the basis of many more complex mathematical concepts.
• Their solutions are easy to understand and implement.

The slope \(m\) tells you how much \(y\) changes when \(x\) increases by 1. Negative slopes tilt the line downwards, while positive slopes tilt it upwards. A key aspect of linear equations is their simplicity and predictability, providing a stable foundation from which to explore more intricate mathematical ideas.

Y-Intercept

The y-intercept is a crucial part of the equation of a line. It tells you where the line crosses the y-axis on a graph. In the slope-intercept form \[ y = mx + b \]the y-intercept is represented by \(b\). This point is pivotal because:

• It serves as the starting point of the line on the graph when \(x = 0\).
• It's where the line meets one of the main axes, providing a clear reference point.
• Changing this value moves the line up or down along the y-axis.

In our example with \(b = 0\), the line goes through the origin \((0,0)\), meaning the line crosses right at the point where both axes meet. Knowing the y-intercept allows us to predict the position and orientation of the line before graphing it, making it an essential aspect of understanding linear equations.