Question

1. vocabulary how can you determine the number of solutions of a system by looking at the equations? lesson 5-1

Explanation:

For a system of linear equations (e.g., in two variables, \(ax + by = c\) and \(dx + ey = f\)), we analyze slopes and y - intercepts:

1. If the slopes (\(m_1=\frac{-a}{b}\), \(m_2=\frac{-d}{e}\)) are different (\(m_1

eq m_2\)), the lines intersect at one point, so the system has one solution.

1. If the slopes are equal (\(m_1 = m_2\)) and the y - intercepts (\(b_1=\frac{c}{b}\), \(b_2=\frac{f}{e}\)) are also equal (\(b_1=b_2\)), the lines are coincident, and the system has infinitely many solutions.
2. If the slopes are equal (\(m_1 = m_2\)) but the y - intercepts are different (\(b_1

eq b_2\)), the lines are parallel, and the system has no solution.

For non - linear systems, we analyze the nature of the equations (e.g., degree, type of functions like quadratic, exponential). For example, a linear and a quadratic equation can intersect 0, 1, or 2 times depending on their graphs. But for linear systems, the slope - intercept analysis (or equivalent methods like elimination to see the result) is key.

Answer:

For a linear system (e.g., \(ax + by = c\), \(dx + ey = f\)):

• One solution: If slopes (\(m_1=\frac{-a}{b}\), \(m_2=\frac{-d}{e}\)) are unequal (\(m_1

eq m_2\)) (lines intersect).

• Infinitely many solutions: If slopes are equal (\(m_1 = m_2\)) and y - intercepts (\(b_1=\frac{c}{b}\), \(b_2=\frac{f}{e}\)) are equal (\(b_1 = b_2\)) (lines coincide).
• No solution: If slopes are equal (\(m_1 = m_2\)) but y - intercepts are unequal (\(b_1

eq b_2\)) (lines are parallel).

For non - linear systems, analyze the equations’ type (e.g., quadratic, exponential) and their graphical/ algebraic interaction (e.g., a line and parabola may intersect 0–2 times).