Answered

Eliminate $z$ from equations (1) and (2), and equations (2) and (3). What is the resulting system of two linear equations?



(1) $5 x+y+4 z=1$

(2) $3 x+2 y+z=1$

(3) $-x-3 y-z=1$


A$-7 x-7 y=-3, 2 x-y=2$
B$17 x+9 y=5, 2 x-y=0$
C$8 x+3 y=2, 4 x+5 y=2$
D$-7 x-7 y=-3, 4 x+5 y=2$
E$17 x+9 y=5, 4 x+5 y=2$

Verified Answer :

  • Eliminate \(z\) from equations (2) and (3) by adding them, resulting in \(2x - y = 2\).
  • Eliminate \(z\) from equations (1) and (2) by multiplying (2) by 4 and subtracting (1), resulting in \(7x + 7y = 3\), which is equivalent to \(-7x - 7y = -3\).
  • The resulting system of two linear equations in \(x\) and \(y\) is \(-7x - 7y = -3\) and \(2x - y = 2\).
  • Final Answer: \(\boxed{\text{-7x - 7y = -3, 2x - y = 2}}\)

Explanation

  1. Analysis of the System of Equations

We are given the following system of linear equations:
(1) \(5x + y + 4z = 1\)
(2) \(3x + 2y + z = 1\)
(3) \(-x - 3y - z = 1\)

Our goal is to eliminate the variable \(z\) from this system to produce a system of two linear equations in terms of \(x\) and \(y\). We will use the given equations to perform this elimination step by step.

  1. Eliminating \(z\) from Equations (2) and (3)

To eliminate \(z\) from equations (2) and (3), we observe their coefficients for \(z\). In equation (2), the coefficient is \(1\), and in equation (3), the coefficient is \(-1\). By simply adding these two equations, the \(z\) terms will cancel out:
\((3x + 2y + z) + (-x - 3y - z) = 1 + 1\)
Combining the like terms, we get:
\((3x - x) + (2y - 3y) + (z - z) = 2\)
\(2x - y = 2\)
Thus, the first equation in our resulting system is \(2x - y = 2\).

  1. Eliminating \(z\) from Equations (1) and (2)

Next, we eliminate \(z\) from equation (1) and equation (2). Equation (1) has \(4z\), while equation (2) has \(z\). To make the coefficients of \(z\) match, we multiply equation (2) by 4:
\(4(3x + 2y + z) = 4(1)\)
\(12x + 8y + 4z = 4\)

Now, we subtract equation (1) from this result:
\((12x + 8y + 4z) - (5x + y + 4z) = 4 - 1\)
Simplifying this:
\((12x - 5x) + (8y - y) + (4z - 4z) = 3\)
\(7x + 7y = 3\)
Multiplying both sides by \(-1\), we obtain:
\(-7x - 7y = -3\)

  1. Conclusion and Final Result

Comparing our resulting equations \(2x - y = 2\) and \(-7x - 7y = -3\) with the given options, we see they match the first option exactly. Therefore, the resulting system of two linear equations is \(-7x - 7y = -3\) and \(2x - y = 2\). The correct choice is the first one.

Examples

Eliminating variables in a system of linear equations is a fundamental skill in algebra with many real-world applications. For example, if you are managing a small business with three different products (A, B, C) and you have data on total costs and revenues from different time periods or production levels, you can represent these as a system of linear equations. By eliminating one of the variables (like the cost of one product), you can simplify the system to solve for the remaining unknowns, helping you make informed decisions about pricing or production strategies. This algebraic technique allows for efficient problem solving in economics, engineering, and logistics.