Solve the System of Equations x-3y=13 5x+6z=41 2x+4y-z=5

Solution:

Step 1:

Isolate $x$ in the first equation: $x = 13 + 3y$
Substitute this expression for $x$ in the other two equations.

Step 2:

Substitute $x$ in the second equation: $5(13 + 3y) + 6z = 41$

Step 2.1:

Distribute $5$ across $(13 + 3y)$: $65 + 15y + 6z = 41$

Step 2.2:

Substitute $x$ in the third equation: $2(13 + 3y) + 4y - z = 5$

Step 2.3:

Distribute $2$ across $(13 + 3y)$: $26 + 6y + 4y - z = 5$

Step 2.4:

Combine like terms: $26 + 10y - z = 5$

Step 3:

Rearrange the first equation: $x = 3y + 13$

Step 4:

Solve for $y$ in the third equation: $10y - z = -21$

Step 4.1:

Divide the equation by $10$: $y = -\frac{21}{10} + \frac{z}{10}$

Step 5:

Substitute $y$ back into the first and second equations.

Step 5.1:

Substitute $y$ in the first equation: $x = 3(-\frac{21}{10} + \frac{z}{10}) + 13$

Step 5.2:

Simplify the expression for $x$: $x = -\frac{63}{10} + \frac{3z}{10} + 13$

Step 5.3:

Substitute $y$ in the second equation: $65 + 15(-\frac{21}{10} + \frac{z}{10}) + 6z = 41$

Step 5.4:

Simplify the expression: $65 - \frac{63}{2} + \frac{3z}{2} + 6z = 41$

Step 6:

Solve for $z$ in the simplified second equation.

Step 6.1:

Multiply through by $2$: $67 + 15z = 82$

Step 6.2:

Isolate $z$: $15z = 15$

Step 6.3:

Divide by $15$: $z = 1$

Step 7:

Substitute $z = 1$ back into the expressions for $x$ and $y$.

Step 7.1:

Substitute $z$ in the expression for $x$: $x = \frac{3(1)}{10} + \frac{67}{10}$

Step 7.2:

Simplify to find $x$: $x = 7$

Step 7.3:

Substitute $z$ in the expression for $y$: $y = -\frac{21}{10} + \frac{1}{10}$

Step 7.4:

Simplify to find $y$: $y = -2$

Step 8:

The solution to the system of equations is $(x, y, z) = (7, -2, 1)$.

Step 9:

The solution can be presented as:

• Point Form: $(7, -2, 1)$
• Equation Form: $x = 7, y = -2, z = 1$

Solution:"The solution to the system of equations is $(7, -2, 1)$."

Knowledge Notes:

1. Substitution Method: This involves solving one equation for one variable and then substituting that expression into the other equations.

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

3. Combining Like Terms: This involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power.

4. Isolating Variables: This involves moving all terms containing the variable to one side of the equation and all other terms to the opposite side.

5. Systems of Equations: A set of equations with multiple variables. The solution is the set of values that satisfies all equations simultaneously.

6. Fractions in Equations: When dealing with fractions, it's often helpful to work with a common denominator or to clear the fractions by multiplying through by the denominator.

7. Checking Solutions: It's important to substitute the found values back into the original equations to ensure they are true solutions.