Solving Equations - Algebra 2

1. 
2. 

Substitute equation 2. into equation 1.,

so,

Substitute into equation 2:

so, the solution is .

1. 
2. 

Substitute equation 2. into equation 1.,

so,

Substitute into equation 2:

so, the solution is .

(1)

When solving a radical equation the fist step is always to isolate the radical. Subtracting from both sides of equation (1).

Square both sides and expand the right side,

Collect all like-terms on onto one side of the equation and use the quadratic formula to find the roots:

(2)                                                                                                                            Reminder

Recall the general solution for the quadratic equation,

(3)                                                                                                                            Use equation (3) to write the solutions to equation (2) and simplify:

Therefore, the roots to the quadratic equation (2) are:

These represent two possible solutions to equation (1). We must check both of them. This is because one of the steps in solving the original equation involved a squaring operation, which can produce fictitious solutions.

Therefore, the only solution for the for equation (1) is:

(1)

When solving a radical equation the fist step is always to isolate the radical. Subtracting from both sides of equation (1).

Square both sides and expand the right side,

Collect all like-terms on onto one side of the equation and use the quadratic formula to find the roots:

(2)                                                                                                                            Reminder

Recall the general solution for the quadratic equation,

(3)                                                                                                                            Use equation (3) to write the solutions to equation (2) and simplify:

Therefore, the roots to the quadratic equation (2) are:

These represent two possible solutions to equation (1). We must check both of them. This is because one of the steps in solving the original equation involved a squaring operation, which can produce fictitious solutions.

Therefore, the only solution for the for equation (1) is:

Step 1: Multiply first equation by −2 and add the result to the second equation. The result is:

Step 2: Multiply first equation by −3 and add the result to the third equation. The result is:

Step 3: Multiply second equation by −23 and add the result to the third equation. The result is:

Step 4: solve for z.

Step 5: solve for y.

Step 6: solve for x by substituting y=2 and z=1 into the first equation.

Step 1: Multiply first equation by −2 and add the result to the second equation. The result is:

Step 2: Multiply first equation by −3 and add the result to the third equation. The result is:

Step 3: Multiply second equation by −23 and add the result to the third equation. The result is:

Step 4: solve for z.

Step 5: solve for y.

Step 6: solve for x by substituting y=2 and z=1 into the first equation.

Step 1: Multiply first equation by −2 and add the result to the second equation. The result is:

Step 2: Multiply first equation by −3 and add the result to the third equation. The result is:

Step 3: Multiply second equation by −23 and add the result to the third equation. The result is:

Step 4: solve for z.

Step 5: solve for y.

Step 6: solve for x by substituting y=2 and z=1 into the first equation.

Step 1: Multiply first equation by −2 and add the result to the second equation. The result is:

Step 2: Multiply first equation by −3 and add the result to the third equation. The result is:

Step 3: Multiply second equation by −23 and add the result to the third equation. The result is:

Step 4: solve for z.

Step 5: solve for y.

Step 6: solve for x by substituting y=2 and z=1 into the first equation.

Add 9 to both sides:

Divide both sides by 27:

Write the second equation in slope-intercept form

This is the same as the first equation.

The two equations represent the same line, therefore there are an infinite number of solutions.

Add three on both sides.

Multiply by on both sides.

Divide by 24 on both sides.

The answer is:

Distribute the x through the parentheses:

x2 –2x = x2 – 8

Subtract x2 from both sides:

–2x = –8

Divide both sides by –2:

x = 4

Given the equation,

and

Plug in for to the equation,

Solve and simplify.

Multiply both sides by 3:

Distribute:

Subtract from both sides:

Add the terms together, and subtract from both sides:

Divide both sides by :

Simplify:

Plug in the value for .

Simplify

Subtract

On the equation, replace x with 2 and then simplify.

Equation 1:

Equation 2:

Equation 3:

Adding the terms of the first and second equations together will yield .

Then, add that to the third equation so that the y and z terms are eliminated. You will get .

This tells us that x = 1. Plug this x = 1 back into the systems of equations.

Now, we can do the rest of the problem by using the substitution method. We'll take the third equation and use it to solve for y.

Plug this y-equation into the first equation (or second equation; it doesn't matter) to solve for z.

We can use this z value to find y

So the solution set is x = 1, y = 2, and z = –5/3.

rearranges to

and

, so

To solve this problem we can first add to each side of the equation yielding

Then we take the square root of both sides to get

Then we calculate the square root of which is .