Question

question 4 (multiple choice worth 5 points)
(solve systems of equations by substitution mc)
a system of linear equations consists of two equations. the first equation has a slope of $-\frac{1}{4}$ and passes through the ordered pair $(-12, 1)$. the second equation passes through the points $(-4, -1)$ and $(-8, 0)$. solve the system of linear equations
$\bigcirc (1, 4)$
$\bigcirc$ no solution
$\bigcirc$ infinitely many solutions
$\bigcirc (-4, 1)$

Explanation:

Step1: Find first equation's equation

Using point - slope form \(y - y_1=m(x - x_1)\), where \(m =-\frac{1}{4}\), \(x_1=- 12\), \(y_1 = 1\).
\(y - 1=-\frac{1}{4}(x + 12)\)
\(y-1=-\frac{1}{4}x-3\)
\(y=-\frac{1}{4}x - 2\)

Step2: Find second equation's slope

Points \((-4,-1)\) and \((-8,0)\). Slope \(m=\frac{0 - (-1)}{-8-(-4)}=\frac{1}{-4}=-\frac{1}{4}\)

Step3: Find second equation's equation

Using point - slope form with point \((-4,-1)\) and \(m =-\frac{1}{4}\)
\(y+1=-\frac{1}{4}(x + 4)\)
\(y + 1=-\frac{1}{4}x-1\)
\(y=-\frac{1}{4}x-2\)

Step4: Analyze the system

Both equations are \(y =-\frac{1}{4}x-2\), so they are the same line. Thus, there are infinitely many solutions.

Answer:

infinitely many solutions