Problem 37 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The equation in the form \(y = mx + c\) is \(y = 4x - 3\). The slope is 4 and the y-intercept is -3. The graph of the line should start at the y-intercept, (0, -3), and then rise 4 units and move one unit to the right repeatedly to obtain the other points on the line.

Step 1: Rewrite the Given Equation in Slope-Intercept Form

The given equation is \(4x-y-3=0\). We need to rearrange the equation so that 'y' is on one side by itself. By adding 'y' to both sides and subtracting '3' from both sides, one gets \(y = 4x - 3\).

Step 2: Identify the Slope and Y-Intercept

From step 1, we've written the equation in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. We can see that \(m = 4\) and \(c = -3\). So, the slope of the line is 4 and the y-intercept is -3.

Step 3: Graph the Equation

Now we graph the equation. First, plot the y-intercept at the point (0, -3). The slope of 4 means for every increase in 1 unit of 'x', 'y' will increase by 4 units. Hence, starting from the y-intercept, one can go up 4 units and to the right one unit to find another point. Repeat this process to find more points. Once enough points have been plotted, a straight line can be drawn through those points and the graph of the line can be obtained.

Key Concepts

Understanding Linear Equations

Linear equations form the backbone of many mathematical concepts you will encounter. When you hear 'linear equation', you might want to picture a straight line. This is because linear equations are, in essence, mathematical representations of straight lines. The standard form of a linear equation in two variables, usually represented as 'x' and 'y', is written as: \[ ax + by = c \]where 'a', 'b', and 'c' are constants. Linear equations reflect a constant rate of change, which means that the rate of change between 'x' and 'y' remains the same throughout. Such equations can be manipulated into the slope-intercept form, which allows us to easily graph the equation and understand the relationship between the two variables involved. They are widely used in different fields such as physics, engineering, economics, and everyday problem-solving scenarios.

Graphing Linear Equations

Graphing linear equations makes the abstract numbers on paper tangible and easier to understand. Before graphing, it’s useful to convert the equation into the slope-intercept form, which is \[ y = mx + c \]where 'm' denotes the slope, and 'c' is the y-intercept. The y-intercept is where the line crosses the y-axis, and the slope is a measure of how steep the line is. Here's a simple method for graphing:

• Start by identifying the y-intercept in the equation to plot your first point on the graph at (0, c).
• Use the slope 'm' to find the next few points. Remember that slope is rise over run, meaning you move 'm' units up or down (rise) for every 1 unit you move right (run).
• Connect the points with a straight line.

With these steps, you can see how 'x' and 'y' relate to each other visually, giving you a greater insight into their relationship.

Slope and Y-Intercept

The slope and y-intercept are two of the most important components of the slope-intercept form of a linear equation. In the equation \[ y = mx + c \],'m' is the slope, which calculates how steep a line is. A positive slope means the line ascends from left to right, and a negative slope indicates it descends. The larger the absolute value of the slope, the steeper the line becomes.

• To determine the slope from a graph, choose two points. Calculate the change in 'y' (rise) over the change in 'x' (run).
• In the given equation, the slope is 4, meaning for every one unit increase in 'x', 'y' increases by 4 units.

The y-intercept, 'c', is where the line meets the y-axis. In our equation, it's -3, which tells us where the line crosses the y-axis. Understanding these two elements helps in predicting how the line behaves on a graph and how different values will affect the line's position and direction.