Factor each trinomial.

1. x^2 + 6x + 5

A: (x + 1)(x +5)

2. x^2 - 7x - 12

A: (x - 3)(x - 4)

Solve each system by substitution.

3. 2x - y = 1; x - y = -2

A: (3, 5)

all 3 are correct.

Thank you.

you are welcome

To factor a trinomial, you want to find two binomials that when multiplied together give you the original trinomial. Here's the process for factorization:

1. Write down the trinomial in the form ax^2 + bx + c.
2. Determine the two numbers that multiply to give a * c, where a is the coefficient of x^2 and c is the constant term.
3. Find the sum of those two numbers, which is also equal to b, the coefficient of x.
4. Rewrite the middle term using the two numbers found in step 2. It will now be in the form bx = mx + nx.
5. Group the four terms (x^2 term, mx term, nx term, and constant term) and factor by grouping.
6. If possible, factor out any common factors between the pairs of terms.
7. Factor out the greatest common factor (GCF) from each group separately.
8. The two resulting binomials form the factored form of the trinomial.

Let's apply this process to the given trinomials:

1. x^2 + 6x + 5:
- a = 1, b = 6, c = 5
- The numbers that multiply to give 1 * 5 are 1 and 5.
- The sum of these numbers is 1 + 5 = 6, which is equal to b.
- Rewrite the middle term: x^2 + 1x + 5x + 5.
- Group the terms: (x^2 + 1x) + (5x + 5).
- Factor out common factors: x(x + 1) + 5(x + 1).
- Factor out the GCF from each group: (x + 1)(x + 5).
- The factored form of the trinomial is (x + 1)(x + 5).

2. x^2 - 7x - 12:
- a = 1, b = -7, c = -12
- The numbers that multiply to give 1 * -12 are 3 and -4.
- The sum of these numbers is 3 + (-4) = -7, which is equal to b.
- Rewrite the middle term: x^2 + 3x - 4x - 12.
- Group the terms: (x^2 + 3x) + (-4x - 12).
- Factor out common factors: x(x + 3) - 4(x + 3).
- Factor out the GCF from each group: (x + 3)(x - 4).
- The factored form of the trinomial is (x + 3)(x - 4).

To solve a system of equations using substitution:

1. Select one equation in the system and solve it for one variable in terms of the other variable.
2. Substitute the expression found in step 1 into the other equation.
3. Solve the resulting equation for the remaining variable.
4. Substitute the value found in step 3 back into the equation from step 1 to find the value of the first variable.
5. Write the solution as an ordered pair (x, y).

Let's apply this process to the given system:

3. 2x - y = 1; x - y = -2
- Solve the second equation for x: x = y - 2.
- Substitute this expression into the first equation: 2(y - 2) - y = 1.
- Simplify the equation: 2y - 4 - y = 1.
- Combine like terms: y - 4 = 1.
- Add 4 to both sides: y = 5.
- Substitute y = 5 back into the second equation: x - 5 = -2.
- Add 5 to both sides: x = 3.
- The solution to the system is (3, 5).