Find The Value Of B - A If The Graph Of Ax + By = 3 Passes Through The Point (-7,2), And Is Parallel
Understanding the relationship between linear equations and their geometric representations on the Cartesian plane is fundamental in algebra. The problem at hand involves determining the difference between the coefficients B and A in a linear equation, given specific conditions about the line's passage through a point and its parallelism to another line. This comprehensive guide aims to elucidate the methods to find the value of B - A systematically, with clear explanations, relevant formulas, and step-by-step solutions.
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Understanding the Problem Statement
Before delving into calculations, it is essential to fully comprehend what the problem entails:
- The linear equation is given as Ax + By = 3.
- The line passes through a specific point (-7, 2).
- The line in question is parallel to another line (which is not explicitly given but can be inferred from the context).
The goal is to determine the value of B - A based on these conditions.
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Key Concepts and Principles
To approach this problem correctly, a review of relevant mathematical concepts is necessary.
1. Equation of a Line in Standard Form
The standard form for a linear equation in two variables is:
\[
Ax + By = C
\]
where:
- \(A\) and \(B\) are coefficients,
- \(C\) is a constant.
In our case, the equation is:
\[
Ax + By = 3
\]
- This form is convenient for analyzing the line's slope and conditions regarding passing through a specific point.
2. Slope of a Line in Standard Form
The slope \(m\) of a line given by \(Ax + By = C\) is:
\[
m = -\frac{A}{B}
\]
provided \(B \neq 0\).
Why is this important?
- Lines are parallel if and only if their slopes are equal.
- Therefore, identifying the slope allows us to set up conditions for parallelism.
3. Passing Through a Point
To verify whether a line passes through a given point \((x0, y0)\), substitute the point's coordinates into the equation:
\[
A x0 + B y0 = C
\]
If the equality holds, the line passes through the point.
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Step-by-Step Solution Approach
To find the value of \(B - A\), follow these logical steps:
Step 1: Use the point to find the relation between A and B
Since the line passes through \((-7, 2)\), substitute these into the line equation:
\[
A \times (-7) + B \times 2 = 3
\]
which simplifies to:
\[
-7A + 2B = 3
\]
This is our first key equation.
Step 2: Understand the condition of parallelism
The line is parallel to another line, say, \(A' x + B' y = C'\). Since the lines are parallel, their slopes are equal:
\[
-\frac{A}{B} = -\frac{A'}{B'}
\]
In our problem, the reference line is \(A x + B y = 3\), which is the same line we're analyzing. But the phrase "and is parallel" suggests that there is another line, possibly the same form, or perhaps a different line with coefficients proportional to \(A\) and \(B\).
Assumption: The problem is asking for the value of \(B - A\) for lines parallel to the given line \(A x + B y = 3\), passing through the point \((-7, 2)\), possibly implying multiple lines with the same slope.
Alternatively, if the problem intends for us to find the value of \(B - A\) for the given line passing through the point and being parallel to a specific line, then the key is that the line's slope is fixed, and the coefficients \(A\) and \(B\) are proportional.
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Step 3: Express \(B\) in terms of \(A\)
From the equation:
\[
-7A + 2B = 3
\]
solve for \(B\):
\[
2B = 7A + 3
\]
\[
B = \frac{7A + 3}{2}
\]
Now, the expression \(B - A\) becomes:
\[
B - A = \frac{7A + 3}{2} - A
\]
Simplify:
\[
B - A = \frac{7A + 3 - 2A}{2} = \frac{5A + 3}{2}
\]
Thus,
\[
B - A = \frac{5A + 3}{2}
\]
This expression relates \(A\) and \(B - A\). To find a specific numerical value, additional conditions are necessary.
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Additional Conditions and Final Calculation
The problem mentions that the line passes through \((-7, 2)\) and is parallel to another line. To fully determine \(A\), \(B\), and consequently \(B - A\), more info about the second line is needed.
Possible interpretations:
- The line is parallel to the original line \(A x + B y = 3\), which is trivial because it is the same line, so the coefficients are proportional.
- The line is parallel to some other specific line, perhaps given elsewhere. Since no other line is provided, the typical approach is to assume the line is parallel to the one with the same slope.
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Special Case: Assuming the line is parallel to the original line
In this case, the line \(A x + B y = 3\) is itself the line passing through the point, or we are asked to find the difference in coefficients for a line passing through \((-7, 2)\) and parallel to this line.
Since the slope of the original line is:
\[
m = -\frac{A}{B}
\]
and the line passes through \((-7, 2)\):
\[
A \times (-7) + B \times 2 = 3
\]
which we already have.
The key is that any line parallel to this one has coefficients proportional to \(A\) and \(B\). For simplicity, assume the coefficients are scaled by some factor \(k\):
\[
A' = kA, \quad B' = kB
\]
but the problem seems to focus on the original coefficients.
---
Final Calculation of \(B - A\) Under Given Conditions
Recall the relation:
\[
B - A = \frac{5A + 3}{2}
\]
To assign a specific numeric value, choose a convenient value for \(A\):
- If \(A = 0\):
\[
B = \frac{7 \times 0 + 3}{2} = \frac{3}{2}
\]
\[
B - A = \frac{3/2 - 0}{1} = \frac{3}{2}
\]
- If \(A = 1\):
\[
B = \frac{7 \times 1 + 3}{2} = \frac{10}{2} = 5
\]
\[
B - A = 5 - 1 = 4
\]
- If \(A = -1\):
\[
B = \frac{7 \times (-1) + 3}{2} = \frac{-7 + 3}{2} = \frac{-4}{2} = -2
\]
\[
B - A = -2 - (-1) = -2 + 1 = -1
\]
Since the problem does not specify a particular value for \(A\), and the relation is linear, the most general answer is:
\[
\boxed{B - A = \frac{5A + 3}{2}}
\]
which depends on the chosen \(A\).
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Summary and Final Remarks
- The key step was substituting the point \((-7, 2)\) into the line's equation to relate \(A\) and \(B\).
- The relation \(B = \frac{7A + 3}{2}\) was derived.
- The expression for \(B - A\) in terms of \(A\) was found as \(\frac{5A + 3}{2}\).
- Without additional information about the specific line to which the given line is parallel, the most precise answer is the algebraic expression:
- If a specific value of \(A\) is provided, the corresponding \(B - A\) can be computed directly.
Additional Tips for Solving Similar Problems
- Always verify the passing point by substitution.
- Use the slope formula to analyze parallel lines.
- Express unknown coefficients in terms of one variable to simplify calculations.
- When multiple lines are involved, consider ratios of coefficients
