Solve by Substitution y=-3x+7 3x-5=-y

Solution:

Step 1: Substitute the value of $y$ from the first equation into the second equation.

• Begin by taking the expression for $y$ from the first equation $y=-3x+7$ and insert it into the second equation in place of $y$.

Step 1.1: Perform the substitution in the equation $3x-5=-y$.

• The substitution gives us $3x-5=-(-3x+7)$.

• Keep the first equation as is: $y=-3x+7$.

Step 1.2: Execute the simplification of the substituted equation.

Step 1.2.1: Simplify the negative sign in front of the parentheses.

Step 1.2.1.1: Distribute the negative sign through the parentheses.

• Apply the distributive property to get $3x-5=3x+(-1\cdot 7)$.

• Retain the first equation: $y=-3x+7$.

Step 1.2.1.2: Carry out the multiplication.

Step 1.2.1.2.1: Multiply $-3$ by $-1$.

• This results in $3x-5=3x+7$.

• The first equation remains unchanged: $y=-3x+7$.

Step 1.2.1.2.2: Multiply $-1$ by $7$.

• This simplifies to $3x-5=3x-7$.

• The first equation is still $y=-3x+7$.

Step 2: Attempt to solve for $x$ in the equation $3x-5=3x-7$.

Step 2.1: Move all terms involving $x$ to one side of the equation.

Step 2.1.1: Subtract $3x$ from both sides.

• This gives us $3x-5-3x=-7$.

• Keep the first equation as is: $y=-3x+7$.

Step 2.1.2: Combine like terms.

Step 2.1.2.1: Subtract $3x$ from $3x$.

• This results in $0-5=-7$.

• The first equation remains $y=-3x+7$.

Step 2.1.2.2: Subtract $5$ from $0$.

• This simplifies to $-5=-7$.

• The first equation is still $y=-3x+7$.

Step 2.2: Since $-5\ne -7$, we conclude that there is no solution to the system of equations.

• The system has no solution.

• The first equation $y=-3x+7$ does not change.

Step 3: End of the process.

Knowledge Notes:

1. Substitution Method: This method involves replacing one variable in an equation with an expression involving another variable, based on another equation in the system. It is often used when one equation in a system is already solved for one of the variables.

2. Simplifying Expressions: This involves applying arithmetic operations and properties such as the distributive property to combine like terms and simplify expressions.

3. Distributive Property: This property states that $a(b+c)=ab+ac$. It is used to remove parentheses by distributing the multiplication over addition or subtraction inside the parentheses.

4. No Solution: When simplifying and solving a system of equations leads to a false statement such as $-5=-7$, it indicates that there is no set of values for the variables that will satisfy both equations simultaneously. This means the system has no solution.

5. System of Linear Equations: A set of two or more linear equations containing two or more variables. The solution to the system is the set of variable values that satisfies all equations in the system simultaneously.

6. Consistency of a System: A system of equations is consistent if it has at least one solution, and inconsistent if it has no solution. The given system in this problem is inconsistent.