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



Answer :

GinnyAnswer
To determine which statement is true about the given system of equations:

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

we can solve the system of equations step by step.

### Step 1: Rewrite the equations in a simpler form

The second equation can be rewritten by eliminating the fractions. Multiply every term by 12 (the least common multiple of 4 and 3) to clear fractions:

[tex]\[ 12 \left(\frac{1}{4}x - \frac{1}{3}y = 1\right) \][/tex]

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

Now the system of equations looks like this:

1. [tex]\(-3x + 4y = 12\)[/tex]
2. [tex]\(3x - 4y = 12\)[/tex]

### Step 2: Add the equations

Add the two equations together to see if we can eliminate one of the variables:

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

Simplifying the left-hand side:

[tex]\[ (-3x + 3x) + (4y - 4y) = 24 \][/tex]

[tex]\[ 0 + 0 = 24 \][/tex]

This results in:

[tex]\[ 0 = 24 \][/tex]

This is a contradiction, which means the two equations represent parallel lines that never intersect. Hence, there is no solution to this system of equations.

### Conclusion

The correct statement is:

- The system of equations has no solution; the two lines are parallel.

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