Question

BRAINLIEST!!!!!!!!!!!!!!!!!!!!!

Answer 1

Hello there!

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.

I hope I helped! 

I will still trying to see if I can solve them another way that might be clearer.

Answer 2

A) Take the equation 2^x = 4^x - 2.

Write the equation in exponential form.

2^x = (2²)^x - 2

Using (a^m)^n = (a^n)^m, rewrite the equation.

2^x = (2^x)² - 2

Basically this equation states that 2 to the x = 2 to the x squared minus 2. You can use substitution.

Rewrite the equation saying that x = x² - 2.

Solve for x by using the quadratic formula.

Move x to the other side.

0 = x² - x - 2.

After a series of long steps, you would get :

x = 2 and x = -1.

Knowing this information, substitute the numbers into the previous formula : x = 2^x

2 = 2^x and -1 = 2^x

Solve for x.

1) 2^x = 2

2^x = 2¹

Since the bases are the same, the exponents are equal.

x = 1

2) -1 = 2^x

The bases are different signs so there is no solution.

Ultimately, x = 1.

If you look at the graph above, this answer is supported because the graphs intersect at where x = 1.

The point of intersection between the two graphs is (1,2).

B) If you having graphing calculator, you can input the equations 2^x and 4^x - 2 and go to check the tables for each function.

C) Use a graphing calculator to graph both functions individually and see where both functions intersect. The point on intersection is where 2^x = 4^x - 2.