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Which statement is true about the equations [tex]\(-3x + 4y = 12\)[/tex] and [tex]\(\frac{1}{4}x - \frac{1}{3}y = 1\)[/tex]?

A. The system of the equations has exactly one solution at [tex]\((-8, 3)\)[/tex].
B. The system of the equations has exactly one solution at [tex]\((-4, 3)\)[/tex].
C. The system of the equations has no solution; the two lines are parallel.
D. The system of the equations has an infinite number of solutions represented by either equation.

Sagot :

GinnyAnswer
Let's analyze the system of equations given:

[tex]\[ -3x + 4y = 12 \][/tex]
[tex]\[ \frac{1}{4}x - \frac{1}{3}y = 1 \][/tex]

To determine the relationship between these two equations, we need to consider their geometric interpretation, specifically whether they are parallel, intersecting, or coincident (overlapping).

### Step 1: Write Equations in Standard Linear Form
First, let's convert both equations to a standard form:

Original equations:
[tex]\[ -3x + 4y = 12 \][/tex]
[tex]\[ \frac{1}{4}x - \frac{1}{3}y = 1 \][/tex]

Rewriting the second equation to avoid fractions:
[tex]\[ 3 \times \left( \frac{1}{4}x - \frac{1}{3}y \right) = 3 \times 1 \][/tex]
[tex]\[ \frac{3}{4}x - y = 3 \][/tex]

### Step 2: Compare the Slopes
The slopes of the lines from the equations in the form [tex]\( Ax + By = C \)[/tex]:

For the first equation:
[tex]\[ -3x + 4y = 12 \][/tex]
To find the slope, [tex]\( m \)[/tex]:
[tex]\[ y = \frac{3}{4}x + 3 \][/tex]
So, the slope [tex]\( m_1 = \frac{3}{4} \)[/tex].

For the second equation:
[tex]\[ \frac{1}{4}x - \frac{1}{3}y = 1 \][/tex]
Multiplying by 12 to clear the denominators:
[tex]\[ 3x - 4y = 12 \][/tex]

Here, we have exactly the same coefficients as the first equation except for the signs, indicating these lines might be the same line or logically inconsistent.

### Step 3: Check for Intersection
We solve the system of equations by either substitution or elimination method:

Let's solve the system:

[tex]\[ -3x + 4y = 12 \][/tex]
[tex]\[ 3x - 4y = 12 \][/tex]

Adding these equations:
[tex]\[ -3x + 4y + 3x - 4y = 12 + 12 \][/tex]
[tex]\[ 0 = 24 \][/tex]

This yields an inconsistent statement, hence the system of equations has no solution.

### Step 4: Conclusion
Given that the lines yield an impossible statement when adding them together, they do not intersect and they are not identical lines. Therefore, they must be parallel lines.

The correct statement about the system of equations is:
- The system of the equations has no solution; the two lines are parallel.

Hence, the true statement regarding the given system of equations is:
[tex]\[ \boxed{\text{The system of the equations has no solution; the two lines are parallel.}} \][/tex]
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