Question

Solve the system of equations. $$\left\\{\begin{aligned} 6 y+4 z &=-12 \\ 3 x+3 y &=9 \\ 2 x-3 z &=10 \end{aligned}\right.$$

Short answer

The solution to the system of equations are \(x = 2, y = 1, z = -1\).

Step-by-step solution

Rearranging Terms in First Equation

Initially, divide the second equation by 3 to simplify it i.e. \(x+y=3\). Thus, you can attain \(y\).Next, from the first equation, express \(z\) in terms of \(y\) by rearranging terms: \(z = -(\frac{6y+12}{4}) = -(1.5y + 3)\). This will allow eliminating \(z\) later on.

Substitute \(y\) and \(z\) into Third Equation

Now, express \(y\) in terms of \(x\) from the second equation: \(y = 3 - x\). Replace \(y\) and \(z\) in the third equation as mentioned: \(2x - 3[-(1.5(3-x) + 3)] = 10\).

Solving for \(x\)

Calculation on the above equation: \(2x - 3[-(4.5 - 1.5x) + 3] = 10\) simplifies to \(2x - 3[-1.5x + 3] = 10\), which further simplifies to \(2x + 4.5x - 9 = 10\). Solving with respect to \(x\) yields \(x = 2\).

Solving for \(y\) and \(z\)

Substitute \(x = 2\) in the second equation, \(x + y = 3\). You attain the value of \(y = 1\). Replace \(y = 1\) in the first equation, where you find \(z = -1\).

Key concepts

Linear Equations

Linear equations are among the simplest types of algebraic expressions, making them a cornerstone in algebra.
Each equation represents a straight line in either a two-dimensional or multi-dimensional space.
Linear equations usually come in the form of \( ax + by + cz = d \), where \( a, b, c, \) and \( d \) are constants, and \( x, y, \) and \( z \) are variables.

• The main feature of linear equations is that each term is either a constant or the product of a constant and a single variable.
• They do not include exponents, products of variables, or any higher-level terms (like squared or cubed terms).
• Since they form lines, any system involving two linear equations can be visualized as the intersection of two planes, where the solution is essentially the point(s) where these planes intersect.

Linear equations are utilized primarily to model real-world scenarios because they offer a simple yet effective representation of mathematical relationships. Thanks to their straightforward nature, systems of linear equations are quite useful in fields such as economics, physics, engineering, and countless others.

Solving Equations

Solving equations, especially in a system, involves finding values for the variables that make all the equations in the system true.
We typically solve systems of equations to find the points where lines or planes intersect in space.
In our original exercise, the system consisted of three equations and three unknowns, \( x, y, \) and \( z \).

• Start by isolating one variable, as we've seen with expressing \( y \) in terms of \( x \), which simplifies the solving process.
• Substitution is a powerful method where you replace one variable with an expressed term in terms of other variables.
• After substitution, equations simplify, as shown when \( y \) and \( z \) values were replaced in the third equation, allowing the solving of \( x \).

With these strategies, solving a system of equations becomes a structured process. Take one step at a time, isolate variables, substitute, and solve step-by-step to reveal each unknown.

Algebraic Manipulation

Algebraic manipulation is a process used to change the form of an equation while maintaining equality.
This skill is crucial for successfully solving systems of equations, as it allows for isolating variables and reducing complex expressions.

• Rearranging terms involves adding, subtracting, or factoring terms to simplify or solve parts of an equation, as done when we express \( z \) in terms of \( y \).
• Distributing negative signs and constants is another vital step; for example, distributing the \( -(1.5y + 3) \) allowed us to eliminate \( z \).
• Combining like terms reduces the number of terms and helps in simplifying equations further, as seen when terms involving \( x \) were combined to solve for \( x \).

With consistent practice in algebraic manipulation, solving a system of equations becomes more intuitive and less daunting. The key is to keep the equation balanced – what you do to one side must be done to the other.