Problem 38 - Algebra 1: Concepts and Skills

Question

Write the equation in slope-intercept form. Then graph the equation. $$4 x-y-3=0$$

Step-by-Step Solution

Answer

The slope-intercept form of the equation is \(y = 4x - 3\), which has a slope of 4 and y-intercept of -3. The line graph passes through the points (0, -3) and (1, 1).

Step 1: Re-arranging into slope-intercept form

To write the given equation \(4x - y - 3 = 0\) in slope-intercept form \(y = mx + c\), begin by isolating \(y\) on one side of the equation. Doing so gives us \(y = 4x - 3\)

Step 2: Identifying the slope and the y-intercept

From the slope-intercept form of the equation, we can easily identify the slope 'm' and the y-intercept 'c'. In our equation \(y = 4x - 3\), the slope 'm' is 4 and the y-intercept 'c' is -3

Step 3: Graphing the Equation

Begin by plotting the y-intercept at the point (0, -3) on the graph. Since the slope is 4 (which can be written as 4/1), from the y-intercept, move up 4 units (because of the positive slope) and to the right 1 unit to get the next point. Draw the line passing through these points.

Key Concepts

Graphing Equations

Graphing equations is a fundamental skill in understanding how different equations relate to geometric representations. It allows us to visualize the relationship between variables.
In graphing an equation, the first step is to ensure it's in a form that is easy to plot, usually the slope-intercept form, which is \( y = mx + c \).
This form helps in identifying key components like the slope and y-intercept.

• Start by marking the y-intercept on the y-axis, which is where the line will cross.
  
• Use the slope to determine the direction and steepness of the line.
  
• Join the marked points with a straight edge, extending the line on both sides.
  
• This ensures a clear visual of how changes in x affect y.

Graphing helps in understanding the behavior of linear equations and is a vital step in analyzing functions geometrically.

Linear Equations

Linear equations are equations between two variables that produce a straight line when plotted on a graph.
They have the general form \( ax + by + c = 0 \).
To work with them effectively, we often convert them into slope-intercept form, \( y = mx + c \), for easier graphing.
This conversion involves solving for y and rearranging other terms accordingly.

• Linear equations always represent a straight line, with consistency in slope and direction.
  
• Each solution to the equation represents a point on the line.
  
• The slope indicates the rate of change in y with respect to x.

Understanding linear equations is crucial as they serve as the foundation for more complex algebraic and calculus problems.This knowledge is useful in both academic exercises and real-life applications, such as determining rates of change and predicting future values.

Slope and Y-Intercept

Slope and y-intercept are essential components in the equation of a line in slope-intercept form, \( y = mx + c \).
Here, 'm' represents the slope, while 'c' denotes the y-intercept.

• Slope (m): The slope is a measure of the steepness of a line, defined as the change in y divided by the change in x (rise over run).
  In the equation \( y = 4x - 3 \), the slope is 4, meaning for every unit increase in x, y increases by 4 units.
  
• Y-Intercept (c): This is the point where the line crosses the y-axis (x=0).
  In our equation, the y-intercept is -3, indicating the point (0, -3) on the graph.
  

These components are instrumental in graphing linear equations and provide valuable insights into the behavior and direction of lines.Understanding them facilitates the analysis of relationships between variables and enhances problem-solving skills in algebra.