algebra 1 question help answer please!!!!

Question

Part A: Explain why the x-coordinates of the points where the graphs of the equations y = 2-x and y = 4x + 3 intersect are the solutions of the equation 2-x = 4x + 3. (4 points)

Part B: Make tables to find the solution to 2-x = 4x + 3. Take the integer values of x only between -3 and 3. (4 points)

Part C: How can you solve the equation 2-x = 4x + 3 graphically? (2 points)

Answer 1

Part A:  In order to distinguish the two line, make the Y in (formula of) the first line a Y₁ and the other Y₂.

 

Y₁= -x +2

Y₂= 4x +3

 

Now when the two lines intersect the values for the Y's at that point must be equal. So

 

-x +2 = Y₁ = Y₂ = 4x +3

 

Part B: What? Make tables? Doesn't you teacher know algebra?

 

-x +2 = 4x +3

 

Isolate the x's

 

2-3 = 4x +x

-1 = 5x

x = -1/5 = - 0.2  ==> Y₁ = Y₂= -(-0.2) + 2 = 2.2  

 

So the lines should intersect at the point (-0.2, 2.2)

 

Part C: To do this (easily) put both equations in slope-(y)intercept form.  y = mx +b

In this form m= the slope (rise/run) and b= the y-intercept = the point on the y-axis where the line crosses the y-axis.

 

So for Y₁= -x +2 , put a little mark on the vertical (Y) axis at +2, 

 

now from that point, count 2 (unit spaces) to the right, then from there count down 2 spaces and put a little mark.

Now connecting those 2 points draws the first line.

 

For Y₂= 4x+3, find 3 units up on the y-axis (mark), now from there move to the right 3 units,

From there move up 12 spaces and put a mark. I used 3 and 12 because m=4=12/3.  Now connect the two marks to draw the second line.

 

Again, the lines should intersect at the point (-0.2, 2.2) 

 

 

 

Answer 2

A.  We have two lines:  y = 2-x   and   y = 4x+3

Given two simultaneous equations that are both required to be true.. the solution is the points where the lines cross...  Which is where the two equations are equal..  Thus the solution that works for both equations is when

2-x = 4x+3

because where that is true is where the two lines will cross and that is the common point that satisfies both equations.

 

B.  2-x = 4x+3

 

x     2-x      4x+3
______________

-3     5       -9
-2     4       -5
-1     3       -1
 0     2        3
 1     1        7
 2     0      11
 3    -1      15

The table shows that none of the integers from [-3,3] work because in no case does
2-x = 4x+3

 

To find the solution we need to rearrange the equation to the form x=n

2-x = 4x+3

2 -x + x = 4x + x +3

2 = 5x + 3

2-3 = 5x +3-3

5x = -1

x = -1/5

 

The only point that satisfies both equations is where x = -1/5

Find y:   y = 2-x  = 2 - (-1/5) = 2 + 1/5 = 10/5 + 1/5 = 11/5

Verify we get the same in the other equation

y = 4x + 3   =  4(-1/5) + 3 = -4/5 + 15/5 = 11/5

 

Thus the only actual solution, being the point where the lines cross, is the point (-1/5, 11/5)

 

C.  To solve graphically  2-x=4x+3

we would graph both lines...   y = 2-x  and   y = 4x+3

The point on the graph where the lines cross is the solution to the system of equations ...

[It should be, as shown above, the point (-1/5, 11/5)]

 

To graph y = 2-x   make a table....

We have already done this in part B

 

x     2-x                x    4x+3

_______              ________

-1    3                   -1      -1

 0    2                    0       3

 1    1                    1       7

 

Just graph the points on a cartesian coordinate system and draw the two lines.  The solution is, as stated, the point where the two lines cross on the graph.