Solve the System of Equations x+y=3 x-3y=-9

Solution:

Step 1:

Isolate $x$ in the first equation: $x = 3 - y$.

Now, we have $x + y = 3$ and $x - 3y = -9$.

Step 2:

Substitute $3 - y$ for $x$ in the second equation.

Step 2.1:

Insert $3 - y$ into $x - 3y = -9$: $(3 - y) - 3y = -9$.

Keep $x = 3 - y$ for later use.

Step 2.2:

Combine like terms on the left-hand side.

Step 2.2.1:

Combine $-y$ and $-3y$ to get $3 - 4y = -9$.

Retain $x = 3 - y$ for substitution later.

Step 3:

Determine the value of $y$ from $3 - 4y = -9$.

Step 3.1:

Shift the constant term to the opposite side.

Step 3.1.1:

Subtract $3$ from both sides: $-4y = -9 - 3$.

Keep $x = 3 - y$ for substitution.

Step 3.1.2:

Calculate $-9 - 3$ to get $-4y = -12$.

Maintain $x = 3 - y$ for the next steps.

Step 3.2:

Divide the equation by $-4$ to isolate $y$.

Step 3.2.1:

Divide $-4y$ and $-12$ by $-4$: $y = \frac{-12}{-4}$.

Continue with $x = 3 - y$.

Step 3.2.2:

Simplify the equation to find $y$.

Step 3.2.2.1:

Reduce the fraction by canceling out $-4$: $y = \frac{-12}{-4}$.

Keep $x = 3 - y$ for substitution.

Step 3.2.3:

Finalize the value of $y$.

Step 3.2.3.1:

Divide $-12$ by $-4$ to find $y = 3$.

Retain $x = 3 - y$ to find $x$.

Step 4:

Substitute $y = 3$ back into $x = 3 - y$ to find $x$.

Step 4.1:

Replace $y$ with $3$ in $x = 3 - y$: $x = 3 - 3$.

Keep $y = 3$ as found.

Step 4.2:

Simplify the equation to solve for $x$.

Step 4.2.1:

Calculate $3 - 3$ to get $x = 0$.

Retain $y = 3$ as the solution for $y$.

Step 5:

Combine the values of $x$ and $y$ to form the solution set.

The solution to the system is the ordered pair $(0, 3)$.

Step 6:

Express the solution in different formats.

Point Form: $(0, 3)$ Equation Form: $x = 0, y = 3$

Step 7:

There are no further steps; the solution is complete.

Knowledge Notes:

To solve a system of linear equations, one can use substitution or elimination methods. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can then be solved. Once the value of one variable is found, it is substituted back into one of the original equations to find the value of the other variable.

In the given problem, the substitution method is used. The steps involve isolating one variable, substituting it into the other equation, simplifying, solving for the remaining variable, and then substituting back to find the value of the first variable.

When simplifying equations, it is essential to combine like terms and perform operations such as addition, subtraction, multiplication, and division to isolate the variables. The goal is to find a solution that satisfies both original equations, which will be an ordered pair representing the point of intersection of the two lines represented by the equations.

In LaTeX, equations are formatted using markup commands to display mathematical expressions clearly. For example, fractions are written using the \frac{}{} command, and variables are often italicized to distinguish them from regular text.