Slope-Intercept Form: Videos & Practice Problems

Understanding linear equations involves mastering the slope-intercept form $y=mx+b$, where m is the slope (rise over run) and b is the y-intercept, the point where the line crosses the y-axis. When given a slope and a point not on the y-axis, the point-slope form $y-y_1=m(x-x_1)$ is used. Both forms describe the same line and can be converted between each other, aiding in graphing and understanding linear relationships.

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is essential for graphing and analyzing lines. The slope-intercept form is expressed as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) denotes the y-intercept. The slope indicates how steep the line is, calculated as the ratio of the rise (change in y) to the run (change in x), often represented as \(m = \frac{\Delta y}{\Delta x}\).

To find the y-intercept, you simply identify the point where the line crosses the y-axis, which occurs when \(x = 0\). For example, if a line crosses the y-axis at \(y = 3\), then \(b = 3\). If the line crosses at \(y = -3\), then \(b = -3\).

For instance, if a line has a slope of \(m = 2\) and a y-intercept of \(b = 3\), the equation in slope-intercept form would be \(y = 2x + 3\). If the slope is \(m = 1\) and the y-intercept is \(b = -3\), the equation simplifies to \(y = x - 3\).

In summary, to write the equation of a line in slope-intercept form, determine the slope and the y-intercept from the graph, and substitute these values into the equation \(y = mx + b\). This method provides a clear and concise way to represent linear relationships in mathematics.

In the graph shown, identify the $y$–intercept & slope. Write the equation of this line in Slope-Intercept form.

• A. $y=\frac23x+1$
• B. $y=-\frac23x+1$
• C. $y=-2x+1$
• D. $y=x+2$

Answer: (B)

Determine the equation of a line given its slope and $y$-intercept. Write in slope-intercept form.
Slope = $4$
$y$-intercept = $-2$

• A. $y=-2x+4$
• B. $y=4x+2$
• C. $y=4x-2$
• D. $y=2x-4$

Answer: (C)

Determine the equation of a line given its slope and $y$-intercept. Write in slope-intercept form.
Slope = $-3$
$y$-intercept = $\frac{5}{2}$

• A. $y=-3x+\frac52$
• B. $y=\frac52x-3$
• C. $y=\frac52x+3$
• D. $y=3x-\frac52$

Answer: (A)

Slope-Intercept Form Example 1

Understanding how to identify the slope of a line from its equation is a fundamental skill in algebra and coordinate geometry. The slope represents the rate of change of the line, often described as "rise over run," and is crucial for analyzing linear relationships.

Consider the equation y = -3. At first glance, it might seem challenging to determine the slope because the equation does not explicitly show the variable x. However, this equation can be rewritten in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. By expressing y = -3 as y = 0x - 3, it becomes clear that the slope m is 0. This means the line is horizontal, running parallel to the x-axis, and there is no vertical change as you move along the line. Graphically, this line crosses the y-axis at -3 and remains constant, confirming that the slope is zero.

On the other hand, the equation x = 5 represents a vertical line crossing the x-axis at 5. Unlike horizontal lines, vertical lines do not have a defined slope because the "run" (horizontal change) is zero, and division by zero is undefined in mathematics. Since slope is calculated as rise over run, the slope of a vertical line is considered undefined. This distinction is important because it highlights that not all lines can be described by a finite slope value.

In summary, horizontal lines have a slope of zero, indicating no vertical change, while vertical lines have an undefined slope due to zero horizontal change. Recognizing these unique cases helps in graphing lines accurately and understanding their behavior in various mathematical contexts. Whether using slope-intercept form, rise over run, or graphing techniques, these strategies enable you to determine the slope effectively for any linear equation.

Graphing Lines in Slope-Intercept Form

Understanding the different forms of linear equations is crucial for graphing and analyzing lines. The slope-intercept form of a line is expressed as y = mx + b, where m represents the slope and b is the y-intercept. This form is particularly useful when you have the slope and y-intercept readily available, allowing for quick graphing. For instance, in the equation y = \frac{2}{3}x - 1, the slope is \frac{2}{3} and the line crosses the y-axis at (0, -1).

However, there are situations where you may only know the slope and a specific point through which the line passes. In such cases, the point-slope form of a line is more appropriate. The point-slope form is given by the equation y - y_1 = m(x - x_1), where (x_1, y_1) is a point on the line and m is the slope. For example, if a line has a slope of \frac{2}{3} and passes through the point (3, 1), you can substitute these values into the point-slope form to get y - 1 = \frac{2}{3}(x - 3).

To graph the line using the point-slope form, start at the point (3, 1). From there, apply the slope of \frac{2}{3} by moving up 2 units and right 3 units to find another point on the line. Alternatively, you can move down 2 units and left 3 units to find a point in the opposite direction. Connecting these points will yield the graph of the line.

It’s important to note that both the slope-intercept form and the point-slope form can represent the same line, just expressed differently. To convert from point-slope form back to slope-intercept form, you can distribute and isolate y. For instance, starting with y - 1 = \frac{2}{3}(x - 3), distributing gives y - 1 = \frac{2}{3}x - 2. Adding 1 to both sides results in y = \frac{2}{3}x - 1, which is the slope-intercept form.

In summary, both forms are essential tools in algebra for representing linear relationships, and understanding when to use each can greatly enhance your ability to work with equations and graphs of lines.

Identify the $y-$intercept & slope of $y=-2x-3$. Then graph the equation.

• A. $b=-2,m=-3$
• B. $b=-3,m=-2$
• C. $b=\frac23,m=-3$
• D. $b=2,m=-3$

Answer: (B)

Graph the line represented by the equation $y=-\frac{4}{5}x+2$

• A.
• B.
• C.
• D.

Answer: (C)

Graphing in Slope-Intercept Form Example 2

The weekly earnings of a rideshare driver can be modeled by the linear equation y = 18.5x − 72.5, where y represents the net weekly profit in dollars, and x is the number of rides given. This equation is in the slope-intercept form y = mx + b, where the slope m = 18.5 indicates the profit earned per ride, and the y-intercept b = −72.5 represents the fixed loss when no rides are given.

To find the driver's profit when no rides are given, substitute x = 0 into the equation. This yields:

\[y = 18.5 \times 0 - 72.5 = -72.5\]

This means the driver incurs a loss of \$72.50 for the week without any rides, reflecting fixed costs or expenses that must be covered regardless of activity.

When the driver completes 15 rides, substituting x = 15 gives:

\[y = 18.5 \times 15 - 72.5 = 277.5 - 72.5 = 205\]

Thus, the net profit for 15 rides is \$205, showing how earnings increase with the number of rides.

Interpreting the components of the equation, the slope of 18.5 represents the incremental profit earned per ride, meaning each additional ride adds \$18.50 to the weekly profit. The y-intercept of −72.5 indicates the baseline loss when no rides are provided, which could be due to fixed costs such as vehicle maintenance or platform fees.

This linear model effectively demonstrates how a rideshare driver's weekly profit depends on the number of rides given, highlighting the importance of completing enough rides to overcome fixed losses and achieve positive earnings. Understanding the slope and intercept in this context helps connect algebraic concepts to real-world financial scenarios.

Here’s what students ask on this topic:

What is the slope-intercept form of a linear equation and how do you identify its components?

The slope-intercept form of a linear equation is given by the MathML equation: $y=mx+b$. In this equation, $m$ represents the slope of the line, which measures the steepness and direction, calculated as the rise over run (change in y over change in x). The $b$ is the y-intercept, the point where the line crosses the y-axis (where $x=0$). Understanding these components allows you to quickly graph the line or write its equation from a graph by identifying the slope and the y-intercept.

How do you write the equation of a line in slope-intercept form when given a graph?

To write the equation of a line in slope-intercept form from a graph, first identify the y-intercept ($b$) by locating where the line crosses the y-axis. This value is the constant term in the equation. Next, determine the slope ($m$) by choosing two points on the line and calculating the rise over run: $m=\frac{\mathrm{\Delta y}}{\mathrm{\Delta x}}$. Once you have $m$ and $b$, plug them into the formula $y=mx+b$ to get the equation of the line.

What is the difference between slope-intercept form and point-slope form of a line?

The slope-intercept form is $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept. It is used when you know the slope and the y-intercept of the line. The point-slope form is $y-y_1=m(x-x_1)$, where $m$ is the slope and $x_1,y_1$ is a point on the line. This form is used when you know the slope and a point on the line that is not necessarily the y-intercept. Both forms describe the same line and can be converted into each other by algebraic manipulation.

How do you convert an equation from point-slope form to slope-intercept form?

To convert from point-slope form $y-y_1=m(x-x_1)$ to slope-intercept form $y=mx+b$, follow these steps: First, distribute the slope $m$ across the parentheses: $y-y_1=mx-m{x}_1$. Then, add $y_1$ to both sides to isolate $y$: $y=mx-m{x}_1+y_1$. The expression $-m{x}_1+y_1$ is the y-intercept $b$ in slope-intercept form.

How do you graph a line using the slope-intercept form?

To graph a line using the slope-intercept form $y=mx+b$, start by plotting the y-intercept $b$ on the y-axis. This is the point where the line crosses the y-axis (at $x=0$). Next, use the slope $m$, which is the ratio of rise over run, to find another point. For example, if the slope is $\frac{2}{3}$, from the y-intercept, move up 2 units and right 3 units to plot the second point. Draw a straight line through these two points to complete the graph.

When should you use point-slope form instead of slope-intercept form?

You should use the point-slope form $y-y_1=m(x-x_1)$ when you are given the slope of a line and a point on the line that is not the y-intercept. This form is especially useful when the point provided is any coordinate $(x_1,y_1)$ other than where the line crosses the y-axis. In contrast, slope-intercept form is more convenient when you know the slope and the y-intercept directly. Both forms describe the same line and can be converted between each other.