# Algebra 1 : Systems of Equations

## Example Questions

### Example Question #16 : How To Find The Solution To A Quadratic Equation

Explanation:

FOIL (first, outside, inside, last) is a device to help students remember to multiply every term by every other term exactly one time. The steps of using FOIL progress as follows:

1) Multiply the first term of each parenthetical expression together:

2) Next, multiply the "outside" terms together:

Add that to the first product, since the terms are all being added together in the parentheses, and the distributive property requires that we maintain the sign:

3) Multiply the "inside" terms together:

Add that to the first two products:

4) Finally, multiply the last terms from each parenthetical expression together:

Add all of the terms together and combine like terms where possible:

5) We now have a quadratic equation. In order to solve a quadratic, we must first set our expression equal to zero by subtracting 6 from both sides.

6) Next, we factor by using backwards distribution. The idea is to split up the middle term, , in such a way that one part has a common factor with  and the other part has a common factor with 6. We can then factor by grouping. Obviously, no matter how we split up , one part will always contain an . So we must focus our attention on the 6. What are the factors of 6 that add up to 7?

Yes!

So,

7) Now we pull out the common factor from each pair of terms,  from the first and 6 from the second:

8) Now we factor again, taking out the common factor  from each term:

9) We know that at least one of the terms must be equal to zero in order for the product to equal zero, so we set each part equal to zero and solve:

and

### Example Question #71 : Systems Of Equations

Where do the graphs of these two equations intersect?

Equation 1:

Equation 2:

Explanation:

One can find the points of intersection of these two functions by setting them equal to one another, essentially substituting  in one equation for the -side of the other equation. This will tell us when the  (output) will be the same in each equation for a given  (input).

By simplifying this equation and setting it equal to zero, we can find the two -values that produce the same  values in the system of the two equations.

Subtract from both sides of the equation, and add to both sides.

Factoring this last equation makes it easier to find the -values that will result in zero on the left side of the equation. Set the two parenthetical phrases equal to zero to find two separate -values that satisfy the equation. These  values will be the  values of the points of intersection between the two equations.

We know our factors multiply to , and the six times one factor plus the other is equal to

and , so and are our factors.

Substituting these two values into either of the two original equations results in the -values of the points of intersection.

are the points of intersection.

### Example Question #3 : Finding Roots

Solve for x.

x = 5

x = –5, –2

x = 5, 2

x = 4, 3

x = –4, –3

x = 5, 2

Explanation:

1) Split up the middle term so that factoring by grouping is possible.

Factors of 10 include:

1 * 10= 10    1 + 10 = 11

2 * 5 =10      2 + 5 = 7

–2 * –5 = 10    –2 + –5 = –7 Good!

2) Now factor by grouping, pulling "x" out of the first pair and "-5" out of the second.

3) Now pull out the common factor, the "(x-2)," from both terms.

4) Set both terms equal to zero to find the possible roots and solve using inverse operations.

x – 5 = 0,  x = 5

x – 2 = 0, x = 2

### Example Question #4 : Finding Roots

Solve for x.

x = 2

x = –4, 4

x = –4

x = –5, –2

x = 5, 2

x = –4

Explanation:

1) First step of solving any equation: combine like terms. With quadratics, the easiest step to take is to set the expression equal to zero.

2) There are two ways to do this problem. The first and most intuitive method is standard factoring.

16 + 1 = 17

8 + 2 = 10

4 + 4 = 8

3) Then follow the usual steps, pulling out the common factor from both pairs, "x" from the first and "4" from the second.

4) Pull out the "(x+4)" to wind up with:

5) Set each term equal to zero.

x + 4 = 0, x = –4

But there's a shortcut! Assuming the terms are arranged by descending degree (i.e., ), and the third term is both a perfect square whose square root is equal to half of the middle term, mathematicians use a little trick. In this case, the square root of 16 is 4. 4 * 2=8, so the trick will work. Take the square root of the first and last term, then stick a plus sign in between them and square the parentheses.

And x, once again, is equal to –4.

### Example Question #72 : Systems Of Equations

Solve for :

Explanation:

To factor, find two numbers that sum to 5 and multiply to 6.

Check the possible factors of 6:

1 * 6 = 6

1 + 6 = 7, so these don't work.

2 * 3 = 6

2 + 3 = 5, so these work!

Next, pull out the common factors of the first two terms and then the second two terms:

Set both expressions equal to 0 and solve:

and

### Example Question #71 : Systems Of Equations

Solve for .

Explanation:

Notice that both terms have in them. Factor this out:

Set both factors equal to zero and solve for .

### Example Question #72 : Systems Of Equations

Solve for

Explanation:

First, we want to use the quadratic formula for this equation, to find the values for x:

When the quadratic formula is simplified, it should look like this:

This provides us with 2 possible values for x:

and/or

To confirm whether or not these are values of X, we need to plug them in, to see if the equation makes sense:

12+24-36 = 0, x = 2

108-72-36 = 0, x = -6

Therefore,

x = 2 and x = -6

### Example Question #201 : Equations / Inequalities

Solve for :

Explanation:

We know that this equation can be simplified to the following format:

(X+_)(X+_) = 0

X will be cross-multiplied by X to form the  of the equation. We need to plug in number values that will result in 4x and 3 when cross-multiplied:

(X+3) = 0, X = -3

(X+1) = 0, X = -1

### Example Question #202 : Equations / Inequalities

Solve for :

Explanation:

When looking at the equation, it is clear that both coefficients and the number value are divisible by 5:

So our first step is to divide both sides of the equation by 5:

When further simplified, we get:

When we plug in the X-Value, we can confirm whether or not this is the correct value for X:

Thus we can confirm that this is the correct value for X.

### Example Question #73 : Systems Of Equations

Solve for  using the Quadratic Formula:

Explanation:

The first step to this quadratic equation is to divide it by the least common denominator, which in this case is 2:

Next, we plug the equation into the Quadratic Formula:

and/or

and/or

We now have 2 possible values for X. The next step is to plug both values in to determine if they are correct values of X:

After confirming both values for X work when plugged into the quadratic formula, we can confirm:

and