Final answer:
To write the linear equation in standard form for a line through (4,3) and (8,2), first calculate the slope (-1/4), then write the equation in point-slope form, convert to slope-intercept form, and finally rearrange to get the standard form: x + 4y = 28.
Step-by-step explanation:
To write a linear equation in standard form for the line that goes through two points, such as (4,3) and (8,2), we need to find the slope of the line and use it to create an equation in slope-intercept form before converting it to standard form.
First, let's find the slope (m) using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. In this case, (x1, y1) = (4,3) and (x2, y2) = (8,2), so m = (2 - 3) / (8 - 4) = -1/4.
Now that we have the slope, we can use the point-slope form to create an equation: y - y1 = m(x - x1). Let's use point (4,3): y - 3 = -1/4(x - 4).
To change this to slope-intercept form, we distribute and simplify: y = -1/4x + 4 + 3, which simplifies to y = -1/4x + 7. To convert to standard form, rearrange the equation to get all terms on one side: 1/4x + y = 7. Multiplying through by 4 to eliminate the fraction gives us the final answer in standard form: x + 4y = 28.