For The Line 6x + 3y = 24, Write The Linear Function And Determine The Slope Of The Line Y = 24 – 6X And The Slope Is + 24 Y = 8 – 2x And The Slope Is – 2 Y= 8 + 2X And The Slope Is + 2 Y = 8 – 2x And The Slope Is + 2
The Correct Answer and Explanation is:
To find the linear function and slope from the given equation:
Given:
6x+3y=246x + 3y = 24
Step 1: Solve for yy in terms of xx (i.e., convert to slope-intercept form: y=mx+by = mx + b)
Start with the original equation: 6x+3y=246x + 3y = 24
Subtract 6x6x from both sides: 3y=−6x+243y = -6x + 24
Divide all terms by 3: y=−2x+8y = -2x + 8
This is the linear function in slope-intercept form, where:
- m=−2m = -2 (the coefficient of xx) is the slope
- b=8b = 8 is the y-intercept
Final Answer:
y=−2x+8 and the slope is −2\boxed{y = -2x + 8 \text{ and the slope is } -2}
Explanation
To find the slope of a line given in standard form, you must first rearrange the equation into the slope-intercept form, y=mx+by = mx + b, where mm represents the slope and bb represents the y-intercept.
The original equation is 6x+3y=246x + 3y = 24, which is in standard form Ax+By=CAx + By = C. To convert this into slope-intercept form, solve for yy:
Subtract 6x6x from both sides to isolate the yy-term: 3y=−6x+243y = -6x + 24
Next, divide every term by 3 to solve for yy: y=−2x+8y = -2x + 8
Now the equation is in the form y=mx+by = mx + b, where the slope m=−2m = -2. This slope tells us that for every unit increase in xx, yy decreases by 2 units, indicating a line that goes downward from left to right.
Among the answer choices, only one correctly matches the function and the slope:
- “Y = 8 – 2X and the slope is -2” is correct.
The other options either miscalculate the slope or the expression for yy. For instance, “Y = 24 – 6X and the slope is +24” is incorrect because the slope would be −6-6, not +24+24, and the equation doesn’t match the rearranged form. Thus, understanding how to isolate yy and identify the slope is essential in linear algebra.
