Problem 6 - Elementary Algebra

Question

Find the equation of each line given the following information. Use the slope- intercept form as the final form of the equation. $$ m=4, \text { the point }(5,2) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = 4x - 18.

Step 1: Substitute given values in slope-intercept equation

Replace y with 2, x with 5, and m with 4 in the slope-intercept equation: $$ 2 = 4(5) + b $$

Step 2: Solve for b

Now, we will solve the equation for b: $$ 2 = 20 + b $$ Subtract 20 from both sides of the equation: $$ b = -18 $$

Step 3: Write the equation of the line in slope-intercept form

Finally, we will write the equation of the line using the values of m and b. The slope m is 4, and the y-intercept b is -18: $$ y = 4x - 18 $$ The equation of the line in slope-intercept form is: $$ y = 4x - 18 $$

Key Concepts

Finding the Equation of a Line

Understanding how to find the equation of a line is a critical skill in algebra. It allows you to describe the relationship between two variables, usually x and y, in a way that shows how they change in relation to each other. The most common way to express this relationship is through the slope-intercept form, which is defined as the equation:
\( y = mx + b \).
In this formula, \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis. To find the equation of a line when given a slope and a point, one can use these two pieces of information to solve for \( b \) and write the full equation.
For the given problem, we already have the slope (\( m = 4 \)) and a point on the line (\( (5,2) \)). By substituting the values into the slope-intercept form, we get an equation with one unknown, \( b \), which we can then solve for. After finding \( b \), we simply rewrite the slope-intercept formula with our actual values to get the line's equation.

Slope and Y-Intercept

The slope of a line, represented by \( m \), tells us how steep the line is and the direction it is going. When given the slope and a point, you have almost all the information you need to write the equation of a line in slope-intercept form.
The y-intercept, represented by \( b \) in the equation \( y = mx + b \), is the point where the line crosses the y-axis, which occurs when \( x = 0 \). This gives you an exact starting point for the line on the coordinate plane.

Crucial Points About Slope:

• A positive slope means the line is increasing.
• A negative slope means the line is decreasing.
• The greater the absolute value of the slope, the steeper the line.
• A slope of 0 represents a horizontal line.

Understanding the Y-Intercept:

• The y-intercept provides a clear visual anchor for the line's position.
• Every line will cross the y-axis at some point unless it is parallel to it.
• Finding the y-intercept often involves substituting known values for x and y and solving for \( b \).

Linear Equations in Algebra

Linear equations form the foundation for much of algebra and represent straight lines on a coordinate plane. They are so named because the variables involved appear only to the first power and are not multiplied together.

Key Features of Linear Equations:

• They create a straight line when graphed.
• They have constant rates of change, which is another term for slope.
• They can often be rearranged into the slope-intercept form: \( y = mx + b \).
• The 'x' and 'y' variables represent points on the Cartesian plane.

When solving linear equations in algebra, you're often finding values for x and y that make the equation true. The slope-intercept form is particularly useful because it allows for quick plotting of the line by providing the starting point (y-intercept) and direction and steepness of the line (slope). Mastering the manipulation of these equations is essential for understanding more complex functions and systems of equations later on.