Alright, let's tackle the problem step-by-step by matching the information on the left with the appropriate equations on the right:
1. First, we have: [tex]\( m=3, b=2 \)[/tex]
- This corresponds to the linear equation [tex]\( y = mx + b \)[/tex].
- Substituting [tex]\( m=3 \)[/tex] and [tex]\( b=2 \)[/tex], the equation becomes [tex]\( y = 3x + 2 \)[/tex].
2. Next, we have: [tex]\( m=3, (1,2) \)[/tex]
- Using the form [tex]\( y = mx + b \)[/tex], where [tex]\( m=3 \)[/tex] and substituting the point [tex]\( (1,2) \)[/tex]:
- We have [tex]\( 2 = 3(1) + b \)[/tex].
- This simplifies to [tex]\( 2 = 3 + b \)[/tex], and solving for [tex]\( b \)[/tex] gives [tex]\( b = -1 \)[/tex].
- Thus, the corresponding equation is [tex]\( y = 3x - 1 \)[/tex].
3. Finally, we have the points: [tex]\( (5,2) \)[/tex] and [tex]\( (6,5) \)[/tex]
- We need to find the linear equation [tex]\( y = mx + b \)[/tex].
- We can use one point, for instance, [tex]\( (5,2) \)[/tex]:
- Substituting [tex]\( (5,2) \)[/tex] and [tex]\( m = 3 \)[/tex]:
- We have [tex]\( 2 = 3(5) + b \)[/tex].
- This simplifies to [tex]\( 2 = 15 + b \)[/tex], and solving for [tex]\( b \)[/tex] gives [tex]\( b = -13 \)[/tex].
- To confirm, we can use the second point [tex]\( (6,5) \)[/tex]:
- Substituting [tex]\( (6,5) \)[/tex] and [tex]\( m = 3 \)[/tex]:
- We have [tex]\( 5 = 3(6) + b \)[/tex].
- This simplifies to [tex]\( 5 = 18 + b \)[/tex], and solving for [tex]\( b \)[/tex] gives [tex]\( b = -13 \)[/tex].
- Thus, the corresponding equation is [tex]\( y = 3x - 13 \)[/tex].
Now, let's look at the remaining equations:
[tex]\[
3x + y = 2 \\
y = 3x - 13 \\
y - 3x = 2 \\
-3x + y = -1
\][/tex]
Out of these, we only need the equations that match the original form [tex]\( y = mx + b \)[/tex] for clarity in matching:
1. For [tex]\( y = 3x + 2 \)[/tex] we have: [tex]\( y = 3x - 13 \)[/tex]
2. For [tex]\( y = 3x - 1 \)[/tex] we have: - This matches with [tex]\( -3x + y = -1 \)[/tex], simplified back to the slope-intercept form.
3. For [tex]\( y = 3x - 13 \)[/tex] we have: [tex]\( 3x + y = 2 \)[/tex] and [tex]\( y - 3x = 2 \)[/tex]
So the final matching is:
1. [tex]\( m=3, b=2 \rightarrow y = 3x + 2 \)[/tex]
2. [tex]\( m=3,(1,2) \rightarrow y = 3x - 1 \)[/tex]
3. [tex]\( (5,2),(6,5) \rightarrow y = 3x - 13 \)[/tex]
4. Equation [tex]\( 3x + y = 2 \)[/tex]
5. Equation [tex]\( y = 3x - 13 \)[/tex]
6. Equation [tex]\( y - 3x = 2 \)[/tex]
7. Equation [tex]\( -3x + y = -1 \)[/tex]
The result looks like this:
1. [tex]\( m=3, b=2 \rightarrow y = 3x + 2 \)[/tex]
2. [tex]\( m=3,(1,2) \rightarrow y = 3x - 1 \)[/tex]
3. [tex]\( (5,2),(6,5) \rightarrow y = 3x - 13 \)[/tex]
4. [tex]\( 3x + y = 2 \)[/tex]
5. [tex]\( y = 3x - 13 \)[/tex] <- Same but confirming the format
6. [tex]\( y - 3x = 2 \)[/tex]
7. [tex]\( -3x + y = -1 \)[/tex]
