Problem 11 - Elementary Algebra

Question

Graph the equations. $$ y=1 x-4 $$

Step-by-Step Solution

Answer

Answer: The slope of the equation $$y = 1x - 4$$ is $$1$$, and the y-intercept is $$-4$$. Two points that can be used to graph this equation are $$(0, -4)$$ and $$(1, -3)$$.

Step 1: Identify the slope and the y-intercept

The given equation is already in the slope-intercept form, $$ y = mx + b $$, where $$m$$ represents the slope and $$b$$ represents the y-intercept. Here, $$ m = 1 $$ and $$ b = -4 $$.

Step 2: Plot the y-intercept

The y-intercept is the point where the line passes through the y-axis. In this case, the y-intercept is $$b = -4$$. So, the line passes through the point $$(0, -4)$$. Plot this point on the graph.

Step 3: Use the slope to find other points on the line

The slope of the line, $$m = 1$$, represents the change in $$y$$ for each unit change in $$x$$. In this case, the slope of the line is $$1$$, which means that for each $$1$$ unit increase in $$x$$, $$y$$ also increases by $$1$$. Pick a point on the graph where the y-intercept was plotted, and then move one unit to the right and one unit up. This leads to the point $$(1, -3)$$. Plot this point on the graph.

Step 4: Draw the line connecting the points

Connect the points $$(0, -4)$$ and $$(1, -3)$$ with a straight line. This line represents the graph of the equation $$ y = 1x - 4 $$. The graph of the equation $$y=1x-4$$ is a straight line passing through the points $$(0,-4)$$ and $$(1,-3)$$.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a simple yet powerful way of expressing relationships between variables. It is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
The slope, \( m \), informs us how steep the line is and in which direction it tilts. A positive slope means the line goes upwards from left to right, whereas a negative slope means it goes downwards.
The y-intercept, \( b \), is the point where the line crosses the y-axis. By using the slope-intercept form, we can easily identify these two critical components.

• The slope \( m \) details the rate of change along the line.
• The y-intercept \( b \) shows where the line starts on the y-axis.

Understanding this form makes plotting and interpreting equations much simpler, as it provides clear guidance on both the positioning and direction of the line.

Graphing

Graphing linear equations is a straightforward process once you have an equation in slope-intercept form. Start by plotting the y-intercept, which is our anchor point on the graph.
Once the y-intercept is on the graph, use the slope to determine the next points. The slope \( m \) indicates how much y changes when x changes by one unit. For example, a slope of 1 means move 1 unit up for every 1 unit right.

• Plot the y-intercept \((0, b)\).
• From this point, use the slope to find subsequent points.
• Connect the points with a straight line.

Always remember, every point on the graph satisfies the equation. This means if you were to pick any point on the line, substituting it into the equation would yield a true statement. This verification step ensures that your graphing is accurate.

Y-Intercept

The y-intercept is a crucial concept in understanding graphing. At its core, it represents the point at which the line crosses the y-axis, where the value of x is always zero.
For our example equation \( y = 1x - 4 \), the y-intercept is -4. This tells us that when x is zero, the value of y is -4, placing the starting point of the line at the coordinate \((0, -4)\).
Why is the y-intercept important? It provides an immediate "starting point" on your graph, much like where a story begins can set the scene for the whole narrative. By identifying the y-intercept first, you ground your graph, making it easier to draw the rest of the line accurately.