Question

a system of equations has 1 solution. if $4x - y = 5$ is one of the equations, which could be the other equation?
$y = -4x + 5$
$-2y = -8x - 10$
$y = 4x - 5$
$2y = 8x - 10$

Explanation:

Step1: Recall system of equations solution conditions

A system of linear equations \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\) has one solution if \(\frac{a_1}{a_2}
eq\frac{b_1}{b_2}\) (they are not parallel, i.e., different slopes). First, rewrite the given equation \(4x - y = 5\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept). Solving \(4x - y = 5\) for \(y\), we get \(y = 4x-5\). So the slope of the given line is \(m_1 = 4\).

Step2: Analyze each option

• Option 1: \(y=-4x + 5\). The slope \(m_2=-4\). Since \(m_1 = 4

eq m_2=-4\), the two lines are not parallel, so the system will have one solution.

• Option 2: \(-2y=-8x - 10\). Divide both sides by - 2: \(y = 4x+5\). The slope \(m_2 = 4\), which is equal to \(m_1 = 4\). The y - intercepts are \(-5\) (for \(y = 4x - 5\)) and \(5\) (for \(y = 4x+5\)), so the lines are parallel (distinct lines), and the system has no solution.
• Option 3: \(y = 4x-5\). This is the same as the given equation (same slope and same y - intercept), so the system has infinitely many solutions.
• Option 4: \(2y=8x - 10\). Divide both sides by 2: \(y = 4x-5\). This is the same as the given equation, so the system has infinitely many solutions.

Answer:

\(y=-4x + 5\) (the first option)