To solve the problem of finding which equation correctly represents the line passing through the points [tex]\( Q(0,1) \)[/tex] and [tex]\( R(2,7) \)[/tex], we need to proceed through several steps:
1. Calculate the Slope of the Line:
The formula for the slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given the points [tex]\( Q(0,1) \)[/tex] and [tex]\( R(2,7) \)[/tex]:
[tex]\[
m = \frac{7 - 1}{2 - 0} = \frac{6}{2} = 3
\][/tex]
Thus, the slope of the line is [tex]\( 3 \)[/tex].
2. Determine the Correct Equation:
Now we need to check each equation given and see which one has the correct slope and passes through the points.
- Option 1: [tex]\( y - 1 = 6x \)[/tex]
- Rearrange to the slope-intercept form, [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = 6x + 1
\][/tex]
- The slope here is [tex]\( 6 \)[/tex], which does not match the calculated slope of [tex]\( 3 \)[/tex].
- Option 2: [tex]\( y - 1 = 3x \)[/tex]
- Rearrange to the slope-intercept form:
[tex]\[
y = 3x + 1
\][/tex]
- The slope here is [tex]\( 3 \)[/tex], which matches our calculated slope, and it passes through [tex]\( Q(0,1) \)[/tex] because plugging [tex]\( x = 0 \)[/tex] gives [tex]\( y = 1 \)[/tex].
- Option 3: [tex]\( y - 7 = 2x - 6 \)[/tex]
- Simplify it:
[tex]\[
y - 7 = 2x - 6 \implies y = 2x + 1
\][/tex]
- The slope here is [tex]\( 2 \)[/tex], which does not match our calculated slope of [tex]\( 3 \)[/tex].
- Option 4: [tex]\( y - 7 = x - 2 \)[/tex]
- Simplify it:
[tex]\[
y - 7 = x - 2 \implies y = x + 5
\][/tex]
- The slope here is [tex]\( 1 \)[/tex], which also does not match our calculated slope of [tex]\( 3 \)[/tex].
3. Conclusion:
Among the given options, only the equation in Option 2, [tex]\( y - 1 = 3x \)[/tex], has the correct slope of [tex]\( 3 \)[/tex] and correctly represents the line passing through the points [tex]\( Q(0,1) \)[/tex] and [tex]\( R(2,7) \)[/tex].
Hence, the correct option is:
[tex]\[ \boxed{y - 1 = 3x} \][/tex]
