How do you solve # 2x – y = 4#, #3x + y = 1# by graphing and classify the system?

Question

Hazel Anglin · Accepted Answer

See a solution process below:

We can locate two points on the line for each equation, graph the line by drawing a line through the points. Formula 1 For #x = 0#: #(2 * 0) - y = 4# #0 - y = 4# #-y = 4# #color(red)(-1) xx -y = color(red)(-1) xx 4# #y = -4# or #(0, -4)# For #x = 2# #(2 * 2) - y = 4# #4 - y = 4# #-color(red)(4) + 4 - y = -color(red)(4) + 4# #0 - y = 0# #-y = 0# #color(red)(-1) xx -y = color(red)(-1) xx 0# #y = 0# or #(2, 0)# graph{[-15, 15, -7.5, 7.5]}(x^2+(y+4)^2-0.05)((x-2)^2+y^2-0.05)(2x-y-4)=0 Formula 2 For #x = 0#: #(3 * 0) + y = 1# #0 + y = 1# #y = 1# or #(0, 1)# For #x = 2# #(3 * 2) + y = 1# #6 + y = 1# #-color(red)(6) + 6 + y = -color(red)(6) + 1# #0 + y = -5# #y = -5# or #(2, -5)# graph{(3x+y-1)(x^2+(y-1)^2-0.05)((x-2)^2+(y+5)^2-0.05)(2x-y-4)=0 [-15, 15, -7.5, 7.5]} Solution We can see the lines cross at #(1, -2)# Consequently: The system is independent because the slopes of the lines differ, and it is consistent because there is at least one solution. graph{(3x+y-1)(-2x-y-4)=0 [-6, 6, -3, 3]}((x-1)^2+(y+2)^2-0.015)

Eli Assouline · Answer

undefined To solve the system of equations $2x - y = 4$ and $3x + y = 1$ by graphing, first rearrange each equation to solve for $y$ in terms of $x$. Then, plot the two lines on the same coordinate plane and identify the point where they intersect. This point represents the solution to the system. Finally, classify the system based on the number of solutions it has.

Eli Assouline · Answer

undefined To solve the system of equations $ 2x - y = 4 $ and $ 3x + y = 1 $ by graphing, follow these steps:

1. Rewrite each equation in slope-intercept form $ y = mx + b $ to graph them easily.

For the first equation $ 2x - y = 4 $:
$$ -y = -2x + 4 $$
$$ y = 2x - 4 $$

For the second equation $ 3x + y = 1 $:
$$ y = -3x + 1 $$

2. Plot the graphs of the two equations on the same coordinate plane.

The graph of $ y = 2x - 4 $ is a line with a slope of 2 and y-intercept of -4. 
The graph of $ y = -3x + 1 $ is a line with a slope of -3 and y-intercept of 1.

3. Determine the point of intersection of the two lines.

The point of intersection represents the solution to the system of equations. In this case, the point of intersection is where the two lines intersect on the graph.

4. Classify the system based on the number of solutions.

If the lines intersect at a single point, the system has one unique solution, which means the equations are consistent and independent. 
If the lines are parallel and do not intersect, the system has no solution, indicating inconsistent equations. 
If the lines coincide (overlap), the system has infinitely many solutions, indicating dependent equations.

After graphing the equations and finding the point of intersection, you can determine the classification of the system based on the number of solutions.