Solve the System of Equations 4x-y=6 2x-y/2=4

Solution:

Step 1: Isolate $x$ in the first equation $4x-y=6$.

• Step 1.1: Move $y$ to the right side by adding it to both sides: $4x=y+6$.

• Step 1.2: Divide the entire equation by $4$ to solve for $x$.

  • Step 1.2.1: Divide each term by $4$: $x=\frac{y}{4}+\frac{6}{4}$.

  • Step 1.2.2: Simplify the fraction $\frac{6}{4}$ by reducing it.

    • Step 1.2.2.1: Reduce the fraction by dividing numerator and denominator by $2$: $x=\frac{y}{4}+\frac{3}{2}$.

Step 2: Substitute the expression for $x$ into the second equation $2x-\frac{y}{2}=4$.

• Step 2.1: Replace $x$ with $\frac{y}{4}+\frac{3}{2}$ in the second equation: $2(\frac{y}{4}+\frac{3}{2})-\frac{y}{2}=4$.

• Step 2.2: Expand and simplify the equation.

  • Step 2.2.1: Apply the distributive property: $2\cdot \frac{y}{4}+2\cdot \frac{3}{2}-\frac{y}{2}=4$.

  • Step 2.2.2: Simplify the terms: $\frac{y}{2}+3-\frac{y}{2}=4$.

  • Step 2.2.3: Cancel out the $\frac{y}{2}$ terms and solve for the constant: $3=4$.

Step 3: Analyze the result.

• Since $3=4$ is a false statement, there is no solution to the system of equations. The system is inconsistent.

Knowledge Notes:

The given problem involves solving a system of linear equations. The process includes manipulating the equations to isolate variables and substituting them into other equations to find a solution. Here are some relevant knowledge points:

1. Isolating Variables: To solve for one variable in terms of others, you can add, subtract, multiply, or divide both sides of the equation by the same number (except zero).

2. Simplifying Fractions: Fractions can be simplified by dividing the numerator and the denominator by their greatest common divisor.

3. Substitution Method: This method involves solving one equation for one variable and then substituting that expression into another equation. This is useful when one equation can be easily solved for one variable.

4. Distributive Property: This property states that $a(b+c)=ab+ac$. It is used to expand expressions and is essential when simplifying equations after substitution.

5. Inconsistent System: A system of equations is inconsistent if there is no set of values for the variables that can satisfy all the equations. In this case, the process led to a false statement ($3=4$), indicating that no solution exists.

6. LaTeX Formatting: In the solution, LaTeX is used to format mathematical expressions. For example, $$\frac{y}{4}+\frac{3}{2}$$renders as a fraction in the output.

Understanding these concepts is crucial for solving systems of linear equations and recognizing when a system has no solution.