Solve the System of Equations x-3y=6 -8x-y=6

Solution:

Step:1

Isolate $x$ in the first equation by adding $3y$ to both sides: $x=6+3y$. Keep the second equation as is: $-8x-y=6$.

Step:2

Substitute the expression for $x$ from the first equation into the second equation.

Step:2.1

In the equation $-8x-y=6$, replace $x$ with $6+3y$: $-8(6+3y)-y=6$.

Step:2.2

Expand and simplify the equation.

Step:2.2.1

Distribute $-8$ across $(6+3y)$: $-8\cdot 6-8\cdot 3y-y=6$.

Step:2.2.2

Perform the multiplication: $-48-24y-y=6$.

Step:2.2.3

Combine like terms: $-48-25y=6$.

Step:3

Solve for $y$ in the equation $-48-25y=6$.

Step:3.1

Isolate the $y$ term by adding $48$ to both sides: $-25y=6+48$.

Step:3.2

Simplify and solve for $y$.

Step:3.2.1

Combine the constants on the right side: $-25y=54$.

Step:3.2.2

Divide both sides by $-25$: $y=\frac{54}{-25}$.

Step:3.2.3

Simplify the fraction: $y=-\frac{54}{25}$.

Step:4

Substitute the value of $y$ back into the first equation to solve for $x$.

Step:4.1

In the equation $x=6+3y$, replace $y$ with $-\frac{54}{25}$: $x=6+3(-\frac{54}{25})$.

Step:4.2

Simplify the expression to find $x$.

Step:4.2.1

Multiply $3$ by $-\frac{54}{25}$: $x=6-\frac{162}{25}$.

Step:4.2.2

Convert $6$ to a fraction with a denominator of $25$: $x=\frac{6\cdot 25}{25}-\frac{162}{25}$.

Step:4.2.3

Combine the fractions: $x=\frac{150-162}{25}$.

Step:4.2.4

Simplify the numerator: $x=\frac{-12}{25}$.

Step:5

The solution set for the system of equations is the pair of values for $x$ and $y$.

Step:6

The solution can be expressed in different formats.

Point Form:

$(x,y)=(-\frac{12}{25},-\frac{54}{25})$

Equation Form:

$x=-\frac{12}{25},y=-\frac{54}{25}$

Knowledge Notes:

1. Substitution Method: This method involves solving one of the equations for one variable and substituting the result into the other equation.

2. Isolation of Variables: To solve for one variable in terms of another, we isolate the variable on one side of the equation.

3. Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses.

4. Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power.

5. Solving Linear Equations: The process of finding the value of the variable that makes the equation true.

6. Fraction Simplification: Reducing fractions to their simplest form by dividing the numerator and denominator by their greatest common factor.

7. 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 satisfy all equations simultaneously.