$ax + by \leq c$ is a linear inequality that represents a region on a graph, where $a$ and $b$ are coefficients, $x$ and $y$ are variables, and $c$ is a constant. This inequality describes all the points $(x, y)$ that satisfy the relationship defined by the equation $ax + by = c$, including those that lie below the line represented by this equation. Understanding this concept is essential for graphing linear inequalities, as it allows us to visualize the solution set that meets the given criteria.
congrats on reading the definition of $ax + by \leq c$. now let's actually learn it.
$ax + by \leq c$ can be graphed by first plotting the boundary line $ax + by = c$, which is dashed if the inequality is strict ($<$ or $>$) and solid if it includes equality ($\leq$ or $\geq$).
To determine which side of the boundary line to shade, you can test a point not on the line (often $(0,0)$) in the original inequality.
This inequality can be used in real-world applications, such as optimizing resources in economics or determining constraints in linear programming.
If either coefficient $a$ or $b$ is zero, the inequality represents either a vertical or horizontal line, respectively.
The solutions to $ax + by \leq c$ form a half-plane on one side of the boundary line, indicating all combinations of $(x, y)$ that satisfy the condition.
Review Questions
How do you graph the linear inequality $ax + by \leq c$, and what steps do you take to determine which area to shade?
To graph the inequality $ax + by \leq c$, start by plotting the boundary line from the equation $ax + by = c$. Determine if this line should be dashed or solid based on whether the inequality is strict or includes equality. Next, choose a test point not on the line (like $(0,0)$) and substitute it into the inequality. If the test point satisfies the inequality, shade the region containing that point; otherwise, shade the opposite side.
Explain how you can apply the concept of $ax + by \leq c$ in real-world scenarios, such as resource allocation.
In real-world scenarios like resource allocation, $ax + by \leq c$ can be used to model constraints where $x$ and $y$ represent different resources or products. The coefficients $a$ and $b$ indicate how much of each resource contributes to a total limit set by $c$. By graphing this inequality, decision-makers can visualize feasible combinations of resources that meet production requirements while adhering to limitations, helping them optimize their output efficiently.
Analyze how changing the coefficients in $ax + by \leq c$ affects its graphical representation and solution set.
Changing coefficients in $ax + by \leq c$ alters both the slope and position of the boundary line on the graph. For instance, increasing coefficient $a$ will steepen the slope of the line, while increasing coefficient $b$ will make it less steep. The constant $c$ shifts the line up or down depending on its value. Each variation affects the feasible region; some changes may expand or contract it, ultimately influencing which combinations of $(x,y)$ satisfy the inequality and leading to different solutions in practical applications.
Related terms
Linear Equation: An equation of the form $y = mx + b$, which represents a straight line on a graph.
Feasible Region: The area on a graph that contains all possible solutions to a system of inequalities.
Boundary Line: The line that represents the equation formed by setting the linear inequality to an equality ($ax + by = c$), which divides the graph into regions.