Solve by Substitution -3x+3y=4 y=x+3

Solution:

Step 1:

Substitute the value of $y$ from the second equation into the first equation.

Step 1.1:

In the equation $-3x+3y=4$, replace $y$ with $x+3$ to get $-3x+3(x+3)=4$. Also, keep $y=x+3$ as is.

Step 1.2:

Begin to simplify the equation on the left-hand side.

Step 1.2.1:

Work on simplifying $-3x+3(x+3)$.

Step 1.2.1.1:

Break down the simplification process term by term.

Step 1.2.1.1.1:

Use the distributive property to expand: $-3x+3x+3\cdot 3=4$ and maintain $y=x+3$.

Step 1.2.1.1.2:

Calculate the product of $3$ and $3$: $-3x+3x+9=4$ and restate $y=x+3$.

Step 1.2.1.2:

Combine like terms in the equation.

Step 1.2.1.2.1:

Add together $-3x$ and $3x$: $0+9=4$ and keep $y=x+3$.

Step 1.2.1.2.2:

Combine $0$ and $9$: $9=4$ and reiterate $y=x+3$.

Step 2:

Recognize that the statement $9=4$ is false, indicating that there is no solution to the system of equations.

Step 3:

Conclude that the system has no solution.

Knowledge Notes:

1. Substitution Method: This method involves replacing one variable with an expression containing the other variable in a system of equations. It is particularly useful when one equation is already solved for one of the variables.

2. Distributive Property: This property states that for any real numbers $a$, $b$, and $c$, the equation $a(b+c)=ab+ac$ holds. It is used to expand expressions where a term is being multiplied by a sum or difference.

3. Simplifying Equations: Simplifying an equation involves combining like terms and performing arithmetic operations to reduce the equation to a simpler form or to solve for a variable.

4. No Solution: When simplifying a system of equations leads to a false statement (such as $9=4$), it indicates that the system has no solution. This means that there are no values for the variables that can satisfy both equations simultaneously.

5. Consistency in Systems of Equations: A system of equations is consistent if it has at least one solution, and inconsistent if it has no solution. The given system is inconsistent because the process of substitution leads to a false statement.