Math Problem Statement
Write the linear equation that gives the rule for this table: x: 5, 6, 7, 8 and y: 20, 24, 28, 32.
Solution
The problem in the image asks you to find the linear equation that fits the given table of values. The table of values is:
\hline x & y \\ \hline 5 & 20 \\ 6 & 24 \\ 7 & 28 \\ 8 & 32 \\ \hline \end{array}$$ This represents a linear relationship between $$x$$ and $$y$$. We can use the **slope-intercept form** of a linear equation, which is: $$y = mx + b$$ Where: - $$m$$ is the slope, - $$b$$ is the y-intercept. ### Step 1: Find the Slope $$m$$ The slope $$m$$ can be calculated using the formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Let's take two points from the table: (5, 20) and (6, 24). $$m = \frac{24 - 20}{6 - 5} = \frac{4}{1} = 4$$ So, the slope $$m = 4$$. ### Step 2: Find the Y-Intercept $$b$$ Now that we know the slope, we can substitute one of the points into the equation $$y = mx + b$$ to find $$b$$. Let’s use the point (5, 20): $$20 = 4(5) + b$$ $$20 = 20 + b \quad \Rightarrow \quad b = 0$$ ### Final Equation Now we can write the equation as: $$y = 4x$$ This is the linear equation that represents the relationship between $$x$$ and $$y$$ in the table. Would you like further details or have any questions? --- Here are five related questions for further practice: 1. How do you find the slope if the points have different spacing between the x-values? 2. What does the slope represent in a real-world context? 3. How can you find the equation if the y-intercept is not zero? 4. How would the equation change if the slope were negative? 5. How can you graph this equation based on the points in the table? **Tip:** Always check multiple pairs of points to ensure the slope is consistent across the table when determining the linear equation!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Equations
Algebra
Slope-Intercept Form
Formulas
y = mx + b
m = (y2 - y1) / (x2 - x1)
Theorems
-
Suitable Grade Level
Grades 8-10