Question

7)
46x - 3
44x + 3
8)
29x + 3
28x + 7
9)
13x + 9
14x + 2
10)
15x - 5
14x + 2
11)
11x + 6
12x + 2
12)
x + 75
x + 115

Explanation:

Step1: Identify the problem type (Geometry - Parallel Lines and Transversals)

These problems involve parallel lines cut by a transversal, so we use properties of parallel lines (e.g., corresponding angles, alternate interior angles, or in some cases, equal segments/angles for congruent or equal measures). For each problem, we set the expressions equal (since the lines are parallel, the angles/segments should be equal).

Step2: Solve Problem 7

We have two expressions \(46x - 3\) and \(44x + 3\). Since the lines are parallel, these should be equal (assuming they are corresponding or alternate interior angles).
Set \(46x - 3 = 44x + 3\)
Subtract \(44x\) from both sides: \(2x - 3 = 3\)
Add 3 to both sides: \(2x = 6\)
Divide by 2: \(x = 3\)

Step3: Solve Problem 8

Expressions: \(29x + 3\) and \(28x + 7\)
Set \(29x + 3 = 28x + 7\)
Subtract \(28x\): \(x + 3 = 7\)
Subtract 3: \(x = 4\)

Step4: Solve Problem 9

Expressions: \(13x + 9\) and \(14x + 2\)
Set \(13x + 9 = 14x + 2\)
Subtract \(13x\): \(9 = x + 2\)
Subtract 2: \(x = 7\)

Step5: Solve Problem 10

Expressions: \(15x - 5\) and \(14x + 2\)
Set \(15x - 5 = 14x + 2\)
Subtract \(14x\): \(x - 5 = 2\)
Add 5: \(x = 7\)

Step6: Solve Problem 11

Expressions: \(11x + 6\) and \(12x + 2\)
Set \(11x + 6 = 12x + 2\)
Subtract \(11x\): \(6 = x + 2\)
Subtract 2: \(x = 4\)

Step7: Solve Problem 12

Expressions: \(x + 75\) and \(x + 115\) Wait, no—wait, maybe they are supplementary? Wait, no, looking at the diagram, maybe they are same - side interior angles? Wait, no, the expressions are \(x + 75\) and \(x + 115\)? Wait, no, maybe I misread. Wait, actually, if they are parallel lines, maybe the angles are supplementary? Wait, no, let's check again. Wait, the problem 12: \(x + 75\) and \(x + 115\)? Wait, no, maybe it's a typo or I missee. Wait, actually, if the lines are parallel, and the angles are same - side interior angles, they should be supplementary. But the expressions are \(x + 75\) and \(x + 115\)? Wait, no, maybe it's \(x + 75\) and another angle. Wait, no, let's re - examine. Wait, maybe the two angles are supplementary. So \((x + 75)+(x + 115)=180\)
Combine like terms: \(2x + 190 = 180\)
Subtract 190: \(2x=-10\)
\(x = - 5\)? That seems odd. Wait, maybe I misidentified the angles. Alternatively, maybe the expressions are equal? But \(x + 75=x + 115\) would imply \(75 = 115\), which is false. So likely, it's a supplementary case. So:

\((x + 75)+(x + 115)=180\)
\(2x+190 = 180\)
\(2x=180 - 190=-10\)
\(x=-5\)

But let's go back to the first few problems which are more straightforward.

(Note: Since the user didn't specify which problem, but we'll present the solutions for each as per the steps above. For example, for problem 7, \(x = 3\); problem 8, \(x = 4\); problem 9, \(x = 7\); problem 10, \(x = 7\); problem 11, \(x = 4\); problem 12, \(x=-5\) (assuming supplementary angles))

Answer:

For Problem 7: \(x = 3\)
For Problem 8: \(x = 4\)
For Problem 9: \(x = 7\)
For Problem 10: \(x = 7\)
For Problem 11: \(x = 4\)
For Problem 12: \(x=-5\) (depending on the angle relationship)