Problem 48 - Algebra 1: Concepts and Skills

Question

Write in standard form the equation of the line that passes through the given point and has the given slope. (Lesson 5.4 ) $$ (-7,-7), m=\frac{1}{2} $$

Step-by-Step Solution

Answer

The equation of the line, in standard form, that passes through the point (-7, -7) and has the slope of \( \frac{1}{2} \) is \( x - 2y = 7 \).

Step 1: Substitute given point and slope in slope-intercept form.

Write the equation of a line in slope-intercept form \( y = mx + b \) substituting \( m = \frac{1}{2} \) and the points \( x = -7, y = -7 \). That results in \( -7 = \frac{1}{2}(-7) + b \).

Step 2: Solve for b, the y-intercept.

Resolve the equation for b. This becomes \( -7 = -\frac{7}{2} + b \), so \( b = -7 + \frac{7}{2} = -\frac{7}{2} \).

Step 3: Write down the slope-intercept form.

Substitute b back into the equation to get the slope-intercept form of the equation: \( y = \frac{1}{2}x - \frac{7}{2} \).

Step 4: Convert into the standard form

Multiply every term in the equation by 2 to avoid fractional coefficients, resulting in \( 2y = x - 7 \). Then, rearrange the equation to get x and y on the same side so it can be written in the standard form of a linear equation \( Ax + By = C \), where A, B, and C are integers and A is nonnegative. Thus, the final equation in the standard form is \( x - 2y = 7 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation of a straight line in a simple format. This format is given by the equation \( y = mx + b \).
The variables in this equation include:

• \( y \): The dependent variable, typically represented on the vertical axis of a graph.
• \( x \): The independent variable, usually on the horizontal axis.
• \( m \): The slope of the line, which indicates its steepness and direction. It is calculated as the change in \( y \) divided by the change in \( x \) (\( \text{slope} = \frac{\Delta y}{\Delta x} \)).
• \( b \): The y-intercept, which is the point where the line crosses the y-axis.

This form is extremely useful when you know the slope and y-intercept of a line, as it allows you to write the equation directly. It helps in quickly graphing the line and understanding how changes in \( x \) affect \( y \).
Using the slope-intercept form, solving real-world problems becomes more intuitive as this format emphasizes the relationship between the slope, intercept, and the variable outcomes.

Standard Form

Standard form is another way to write the equation of a line. It is generally arranged as \( Ax + By = C \), where:

• \( A \), \( B \), and \( C \) are integers.
• \( A \) should be a nonnegative integer, meaning it should not be less than zero.

The standard form is particularly useful for quickly identifying the x and y-intercepts of a line. This is because you can easily set \( y \) to zero to find the x-intercept or set \( x \) to zero to find the y-intercept.
To convert from slope-intercept form to standard form, you typically rearrange the terms to get \( x \) and \( y \) on one side and the constant on the other side. You might also need to multiply the entire equation by an integer to eliminate any fractions.
In the problem's solution, combining terms allowed customization into this format from the slope-intercept form, making it easier to interpret graphically or algebraically.

Linear Equations

Linear equations represent relationships with constant rates of change. This means that any given increase in one variable will result in a proportional increase in the other variable, producing a straight line when graphed.
Linear equations are fundamental in mathematics since they describe a wide variety of real-world phenomena. They show predictable patterns, such as consistent speed, steady growth, or uniform decrease.
There are multiple forms of linear equations, each serving specific applications:

• Slope-Intercept Form: Useful for analyzing and graphing lines quickly when you know the slope and the y-intercept.
• Standard Form: Helpful in identifying intercepts and in cases where algebraic manipulation requires integer coefficients.

Understanding these forms and being able to interchange between them based on what information is available or needed will greatly enhance your problem-solving skills. Recognizing that all these forms ultimately describe the same linear relationship is key to mastering linear equations and applying them successfully in varied scenarios.