Math Problem Statement
Graph each equation and inequality using tables, x and y intercepts, and shading regions.
Solution
The image contains a set of graphing problems related to linear equations and inequalities. Let's go through each of the problems one by one:
1. Graph the equation using a table:
- To graph this, you can create a table of values for and compute the corresponding values for .
Example:
x & y \\ \hline 0 & -1 \quad (\text{from } 4(0) - 1 = -1) \\ 1 & 3 \quad (\text{from } 4(1) - 1 = 3) \\ 2 & 7 \quad (\text{from } 4(2) - 1 = 7) \\ \end{array}$$ Plot these points and connect them to form a straight line. --- **2. Graph the equation using a table:** $$3 = 5x - y$$ Rearranging it into slope-intercept form $$y = 5x - 3$$: - You can similarly create a table of values for $$x$$ and calculate $$y$$. Example: $$\begin{array}{c|c} x & y \\ \hline 0 & -3 \quad (\text{from } y = 5(0) - 3) \\ 1 & 2 \quad (\text{from } y = 5(1) - 3) \\ 2 & 7 \quad (\text{from } y = 5(2) - 3) \\ \end{array}$$ Plot these points and connect them to form a line. --- **3. Graph each equation using the X and Y intercepts:** **(a) $$3y - x = 6$$** - To find the x-intercept, set $$y = 0$$ and solve for $$x$$: $$0 - x = 6 \quad \Rightarrow \quad x = -6$$ - To find the y-intercept, set $$x = 0$$ and solve for $$y$$: $$3y - 0 = 6 \quad \Rightarrow \quad y = 2$$ Plot the points $$(-6, 0)$$ and $$(0, 2)$$, then draw the line through them. **(b) $$2x - 2y = 6$$** - For the x-intercept, set $$y = 0$$: $$2x = 6 \quad \Rightarrow \quad x = 3$$ - For the y-intercept, set $$x = 0$$: $$-2y = 6 \quad \Rightarrow \quad y = -3$$ Plot the points $$(3, 0)$$ and $$(0, -3)$$, and draw the line through them. --- **4. Graph the inequality:** $$y + 1 \geq 2x$$ - First, rewrite the inequality as $$y \geq 2x - 1$$. - To graph the boundary line $$y = 2x - 1$$, plot points as you did for equations. - Then shade the region above the line, as the inequality is $$y \geq 2x - 1$$, which includes the points where $$y$$ is greater than or equal to the line. --- Let me know if you need further explanation or details on any specific part! Here are five related questions for practice: 1. What is the slope of the line $$y = 4x - 1$$? 2. How do you find the slope-intercept form of $$5x - y = 3$$? 3. What are the intercepts of the equation $$3x + 2y = 6$$? 4. How would you graph the inequality $$y \leq -x + 3$$? 5. Can you find the slope and y-intercept of the equation $$2x - y = 5$$? **Tip**: When graphing inequalities, always remember to use a dashed line for strict inequalities (like $$y > 2x$$) and a solid line for non-strict ones (like $$y \geq 2x$$).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Graphing
Linear Equations
Inequalities
Formulas
y = mx + b
x-intercept: set y = 0
y-intercept: set x = 0
Theorems
Slope-intercept form
Graphing inequalities: solid vs. dashed line
Shading regions for inequalities
Suitable Grade Level
Grades 7-10