Solve by Substitution y=-1x-5 4x-8y=4

Solution:

Step 1:

Express $-1x$ as $-x$. We have $y=-x-5$ and $4x-8y=4$.

Step 2:

Substitute $-x-5$ for $y$ in each equation.

Step 2.1:

In the equation $4x-8y=4$, replace $y$ with $-x-5$ to get $4x-8(-x-5)=4$. Keep $y=-x-5$ for later use.

Step 2.2:

Begin simplifying the equation.

Step 2.2.1:

Expand $4x-8(-x-5)$.

Step 2.2.1.1:

Distribute $-8$ across $-x$ and $-5$ to get $4x+8x+40=4$. Retain $y=-x-5$.

Step 2.2.1.2:

Combine like terms $4x$ and $8x$ to obtain $12x+40=4$. Keep $y=-x-5$.

Step 3:

Isolate $x$ in the equation $12x+40=4$.

Step 3.1:

Shift terms not containing $x$ to the opposite side.

Step 3.1.1:

Subtract $40$ from both sides to get $12x=4-40$. Keep $y=-x-5$.

Step 3.1.2:

Compute $4-40$ to find $12x=-36$. Retain $y=-x-5$.

Step 3.2:

Divide the equation $12x=-36$ by $12$ to solve for $x$.

Step 3.2.1:

Divide $12x$ and $-36$ by $12$ to get $\frac{12x}{12}=\frac{-36}{12}$. Keep $y=-x-5$.

Step 3.2.2:

Cancel out the common factor of $12$.

Step 3.2.2.1:

Eliminate the common factor to simplify to $x=\frac{-36}{12}$. Retain $y=-x-5$.

Step 3.2.2.2:

Reduce $x$ to $x=-3$. Keep $y=-x-5$.

Step 4:

Insert $-3$ for $x$ in the remaining equations.

Step 4.1:

In the equation $y=-x-5$, substitute $x$ with $-3$ to get $y=-(-3)-5$. Keep $x=-3$.

Step 4.2:

Simplify the equation.

Step 4.2.1:

Resolve $-(-3)-5$.

Step 4.2.1.1:

Multiply $-1$ by $-3$ to get $y=3-5$. Retain $x=-3$.

Step 4.2.1.2:

Subtract $5$ from $3$ to find $y=-2$. Keep $x=-3$.

Step 5:

The solution set consists of the ordered pair that satisfies both equations: $(-3,-2)$.

Step 6:

The solution can be presented in various formats.

• Point Form: $(-3,-2)$
• Equation Form: $x=-3,y=-2$

Step 7:

(No further steps required)

Knowledge Notes:

1. Substitution Method: A technique used to solve systems of equations where one equation is solved for one variable in terms of the others, and then substituted into the remaining equations.

2. Simplification: The process of reducing expressions to their simplest form, often by combining like terms and applying arithmetic operations.

3. Distributive Property: A property that allows us to multiply a sum by a number by multiplying each addend separately and then sum the products: $a(b+c)=ab+ac$.

4. Isolating Variables: The process of manipulating an equation to get a variable by itself on one side of the equation, often to solve for that variable.

5. System of Equations: A set of two or more equations with the same variables. The solution to the system is the set of values that satisfies all equations simultaneously.

6. Ordered Pair: A pair of numbers used to locate a point on a coordinate plane, typically written in the form $(x,y)$.

7. Latex Format: A typesetting system that is widely used for mathematical and scientific documents, due to its powerful handling of formulas and bibliographies.