Without graphing, decide:
(a)Are the graphs of the equations are identical lines, parallel lines, or lines intersecting at a single point?
(b)How many solutions does the system have?
-$\left\{ \begin{aligned}
3 x - y & = 1 \
x + 4 y & = 22
\end{aligned} \right.$
A)lines intersecting at a single point; one solution
B)parallel lines; no solution
C)identical lines; infinite number of solutions

Question

Quizplus · Accepted Answer

ATo determine the relationship between the two lines and the number of solutions, we can compare their slopes and y-intercepts. The first step is to convert both equations into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.For the first equation, $3x - y = 1$, we solve for y to get $y = 3x - 1$.For the second equation, $x + 4y = 22$, we solve for y to get $y = \frac{22 - x}{4}$ or $y = -\frac{1}{4}x + \frac{22}{4}$.Since the slopes of the two lines ($3$ and $-\frac{1}{4}$) are different, the lines are neither parallel nor identical; they must intersect at a single point. Therefore, the system of equations has exactly one solution.

Without Graphing, Decide: (A)Are the Graphs of the Equations Are {3x−y=1x+4y=22\left\{ \begin{aligned}3 x - y & = 1 \\x + 4 y & = 22\end{aligned} \right.{3x−yx+4y​=1=22​

Without Graphing, Decide:
(A)Are the Graphs of the Equations Are $\left\{ \begin{aligned}3 x - y & = 1 \\x + 4 y & = 22\end{aligned} \right.$