91Ó°ÊÓ

Linear equations are mathematical expressions that represent lines when graphed on a coordinate plane. They consist of variables multiplied by constants and added together. In this context, we have a set of linear equations in three variables: \(x\), \(y\), and \(z\). Each equation has variables with different coefficients:

• \(5x + 2y = 4\)
• \(3x + 4y + 2z = 6\)
• \(7x + 3y + 4z = 29\)

The primary goal is to find values of \(x\), \(y\), and \(z\) that satisfy all these conditions simultaneously. Solutions to linear equations are often found using methods like substitution or elimination.

The substitution method is a technique for solving a system of equations where you solve one equation for one variable and then substitute that expression into the other equation(s). In our exercise, we started with the first equation \(5x + 2y = 4\) and solved for \(x\) in terms of \(y\):

• \(x = \frac{4 - 2y}{5}\)

This expression was then substituted into the remaining equations. The substitution allows us to reduce the number of variables in the equation, making it easier to solve for the others. It effectively transforms the problem into one with fewer unknowns. The result of substituting \(x\) in terms of \(y\) helped us establish two resonable two-variable equations that we could solve more straightforwardly.

After calculating potential solutions, it is crucial to verify them by substituting back into the original set of equations. In our exercise, the values found were \(x = 2\), \(y = -3\), and \(z = 6\). Verification involves checking:

• First equation: \(5(2) + 2(-3) = 10 - 6 = 4\), which is true.
• Second equation: \(3(2) + 4(-3) + 2(6) = 6 - 12 + 12 = 6\), also true.
• Third equation: \(7(2) + 3(-3) + 4(6) = 14 - 9 + 24 = 29\), again true.

Each calculation aligns perfectly with the given equations, confirming that our solutions are correct. Solution verification is a crucial final step, as it ensures that the derived values accurately satisfy all the original equations.

Algebraic manipulation is a fundamental skill used to rearrange equations and expressions, making them simpler to evaluate. In this exercise, we employed several algebraic techniques:

• Solving for one variable in terms of others in a single equation, such as obtaining \(x = \frac{4 - 2y}{5}\) from \(5x + 2y = 4\).
• Substituting expressions from one equation into others to eliminate variables, reducing the complexity of the system.
• Ensuring balance by performing the same operations on both sides of the equation, evidenced by multiplying entire equations to clear denominators.
• Simplifying resulting equations to make them more straightforward to solve, such as turning \(14y + 10z = 18\) into a form that is easier to analyze.

These manipulations are essential for finding the solutions systematically and efficiently, ensuring clarity and direction in solving systems of equations.