Question

1. how many solutions for the system of equations? why? (do not solve it)

5x - 4y = 7
6y + 12 = 6x

Explanation:

Step1: Analyze the system of equations

We have two equations: \(5x - 4y = 7\) and \(6y + 12 = 6x\). Let's rewrite the second equation in standard form (\(Ax + By = C\)).
Starting with \(6y + 12 = 6x\), subtract \(6x\) from both sides and subtract \(12\) from both sides: \(-6x + 6y = -12\). We can simplify this by dividing all terms by \(-6\): \(x - y = 2\), or \(x = y + 2\).

Step2: Check the slopes (or coefficients) of the two original equations

For a linear equation in the form \(Ax + By = C\), the slope \(m = -\frac{A}{B}\).
For the first equation \(5x - 4y = 7\), \(A = 5\), \(B = -4\), so slope \(m_1 = -\frac{5}{-4}=\frac{5}{4}\).
For the second equation (after rewriting as \(6x - 6y = 12\) or simplifying to \(x - y = 2\)), \(A = 1\), \(B = -1\), so slope \(m_2 = -\frac{1}{-1}=1\).

Since \(m_1
eq m_2\) (the slopes are different), the two lines are not parallel and not coincident.

Step3: Determine the number of solutions

In a system of linear equations, if the two lines have different slopes, they are not parallel and will intersect at exactly one point. So the system of equations has exactly one solution.

Answer:

The system of equations \(5x - 4y = 7\) and \(6y + 12 = 6x\) has \(\boldsymbol{1}\) solution because the two linear equations have different slopes (their graphs are non - parallel, non - coincident lines), so they intersect at exactly one point.