Question

how many solutions does the system of equations below have?
$y = -9x - 2$
$y = \frac{10}{9}x - \frac{3}{4}$
no solution
one solution
infinitely many solutions
submit
work it out

Explanation:

Step1: Recall the form of linear equations

The given equations are in the slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the first equation \(y=-9x - 2\), the slope \(m_1=-9\) and the y - intercept \(b_1 = - 2\). For the second equation \(y=\frac{10}{9}x-\frac{3}{4}\), the slope \(m_2=\frac{10}{9}\) and the y - intercept \(b_2=-\frac{3}{4}\).

Step2: Analyze the slopes and y - intercepts

Two linear equations \(y = m_1x + b_1\) and \(y=m_2x + b_2\) have:

• No solution if \(m_1=m_2\) and \(b_1

eq b_2\) (parallel lines).

• Infinitely many solutions if \(m_1 = m_2\) and \(b_1=b_2\) (coincident lines).
• One solution if \(m_1

eq m_2\) (intersecting lines).

Here, \(m_1=-9\) and \(m_2 = \frac{10}{9}\), so \(m_1
eq m_2\). When the slopes of two linear lines are different, the lines intersect at exactly one point. So the system of equations has one solution.

Answer:

one solution (corresponding to the option "one solution")