Question

question 1-2
the system of equations below will have
$4x - 3y = 10$
$4x - 3y = 6$
\bigcirc one unique solution
\bigcirc no solution
\bigcirc infinitely many solutions

Explanation:

Step1: Analyze the equations

The two equations are \(4x - 3y = 10\) and \(4x - 3y = 6\). Let's rewrite them in the slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept).

For the first equation \(4x-3y = 10\), we can solve for \(y\):
\(- 3y=-4x + 10\), then \(y=\frac{4}{3}x-\frac{10}{3}\)

For the second equation \(4x - 3y=6\), we solve for \(y\):
\(-3y=-4x + 6\), then \(y=\frac{4}{3}x - 2\)

Step2: Compare slopes and intercepts

The slope \(m\) of both lines is \(\frac{4}{3}\), which means the two lines are parallel. The y - intercepts are \(-\frac{10}{3}\) and \(- 2\) (since \(-2=-\frac{6}{3}\) and \(-\frac{10}{3}
eq-\frac{6}{3}\)), so the lines are parallel and distinct. Parallel lines that are not coincident (same line) have no intersection points. In terms of a system of linear equations, if two lines are parallel and not the same line, the system has no solution.

We can also use the method of subtracting the two equations. Subtract the second equation from the first equation: \((4x - 3y)-(4x - 3y)=10 - 6\), which simplifies to \(0=4\), a contradiction. When we get a contradiction in a system of linear equations, it means there is no solution.

Answer:

no solution