Write the linear equation that gives the rule for the table with the following values, such that y is on the left side of the equation. x = [40, 43, 46, 49] y = [80, 83, 86, 89]
Understand the Problem
The question asks to find the linear equation (in the form y = mx + b) that represents the relationship between x and y shown in the table. We need to determine the slope (m) and y-intercept (b) based on the given data points, and express the equation with 'y' isolated on the left side.
Answer
$y = x + 40$
Answer for screen readers
$y = x + 40$
Steps to Solve
- Calculate the slope (m)
To find the slope $m$, we use the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Using the first two points (40, 80) and (43, 83): $$ m = \frac{83 - 80}{43 - 40} = \frac{3}{3} = 1 $$
- Find the y-intercept (b)
Using the slope-intercept form of a linear equation $y = mx + b$, we can substitute the slope ($m = 1$) and one of the points, say (40, 80), to solve for $b$: $$ 80 = 1 \cdot 40 + b $$ $$ 80 = 40 + b $$ $$ b = 80 - 40 = 40 $$
- Write the linear equation
Now that we have the slope $m = 1$ and the y-intercept $b = 40$, we can write the linear equation: $$ y = 1x + 40 $$ $$ y = x + 40 $$
$y = x + 40$
More Information
The linear equation $y = x + 40$ describes the relationship between $x$ and $y$ in the given table. For every unit increase in $x$, $y$ also increases by one unit, and when $x$ is 0, $y$ is 40.
Tips
A common mistake is incorrectly calculating the slope or y-intercept. Ensure you use the correct formula for the slope and substitute the values accurately. Also, make sure to verify your equation with all given data points to ensure consistency.
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