Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To eliminate the variable [tex]\( y \)[/tex] from the given system of equations, we can follow these steps:
Given the system of equations:
[tex]\[ \left\{\begin{array}{r} x + y - z = -3 \quad \text{(Equation 1)} \\ 7x - 6y - 3z = -35 \quad \text{(Equation 2)} \\ 2x + y - 4z = -14 \quad \text{(Equation 3)} \end{array}\right. \][/tex]
### Step 1: Eliminate [tex]\( y \)[/tex] from Equations 1 and 2
1. Multiply Equation 1 by 6:
[tex]\[ 6(x + y - z) = 6(-3) \][/tex]
This gives:
[tex]\[ 6x + 6y - 6z = -18 \quad \text{(Equation 4)} \][/tex]
2. Subtract Equation 2 from Equation 4:
[tex]\[ (6x + 6y - 6z) - (7x - 6y - 3z) = -18 - (-35) \][/tex]
Simplifying this:
[tex]\[ 6x + 6y - 6z - 7x + 6y + 3z = -18 + 35 \][/tex]
Combining like terms:
[tex]\[ -x + 12y - 3z = 17 \][/tex]
Multiplying the entire equation by [tex]\(-1\)[/tex] to simplify:
[tex]\[ x - 12y + 3z = -17 \quad \text{(Resulting Equation)} \][/tex]
### Step 2: Eliminate [tex]\( y \)[/tex] from Equations 1 and 3
1. Multiply Equation 1 by 1:
[tex]\[ 1(x + y - z) = 1(-3) \][/tex]
This gives:
[tex]\[ x + y - z = -3 \quad \text{(Equation 5)} \][/tex]
2. Subtract Equation 3 from Equation 5:
[tex]\[ (x + y - z) - (2x + y - 4z) = -3 - (-14) \][/tex]
Simplifying this:
[tex]\[ x + y - z - 2x - y + 4z = -3 + 14 \][/tex]
Combining like terms:
[tex]\[ -x + 3z = 11 \][/tex]
Multiplying the entire equation by [tex]\(-1\)[/tex] to simplify:
[tex]\[ x - 3z = -11 \quad \text{(Resulting Equation)} \][/tex]
### Summary:
- To eliminate [tex]\( y \)[/tex] from Equations 1 and 2, multiply Equation 1 by [tex]\( 6 \)[/tex].
- To eliminate [tex]\( y \)[/tex] from Equations 1 and 3, multiply Equation 1 by [tex]\( 1 \)[/tex].
The resulting equations are:
[tex]\[ \left\{ \begin{array}{l} x - 12y + 3z = -17 \\ x - 3z = -11 \end{array} \right. \][/tex]
Given the system of equations:
[tex]\[ \left\{\begin{array}{r} x + y - z = -3 \quad \text{(Equation 1)} \\ 7x - 6y - 3z = -35 \quad \text{(Equation 2)} \\ 2x + y - 4z = -14 \quad \text{(Equation 3)} \end{array}\right. \][/tex]
### Step 1: Eliminate [tex]\( y \)[/tex] from Equations 1 and 2
1. Multiply Equation 1 by 6:
[tex]\[ 6(x + y - z) = 6(-3) \][/tex]
This gives:
[tex]\[ 6x + 6y - 6z = -18 \quad \text{(Equation 4)} \][/tex]
2. Subtract Equation 2 from Equation 4:
[tex]\[ (6x + 6y - 6z) - (7x - 6y - 3z) = -18 - (-35) \][/tex]
Simplifying this:
[tex]\[ 6x + 6y - 6z - 7x + 6y + 3z = -18 + 35 \][/tex]
Combining like terms:
[tex]\[ -x + 12y - 3z = 17 \][/tex]
Multiplying the entire equation by [tex]\(-1\)[/tex] to simplify:
[tex]\[ x - 12y + 3z = -17 \quad \text{(Resulting Equation)} \][/tex]
### Step 2: Eliminate [tex]\( y \)[/tex] from Equations 1 and 3
1. Multiply Equation 1 by 1:
[tex]\[ 1(x + y - z) = 1(-3) \][/tex]
This gives:
[tex]\[ x + y - z = -3 \quad \text{(Equation 5)} \][/tex]
2. Subtract Equation 3 from Equation 5:
[tex]\[ (x + y - z) - (2x + y - 4z) = -3 - (-14) \][/tex]
Simplifying this:
[tex]\[ x + y - z - 2x - y + 4z = -3 + 14 \][/tex]
Combining like terms:
[tex]\[ -x + 3z = 11 \][/tex]
Multiplying the entire equation by [tex]\(-1\)[/tex] to simplify:
[tex]\[ x - 3z = -11 \quad \text{(Resulting Equation)} \][/tex]
### Summary:
- To eliminate [tex]\( y \)[/tex] from Equations 1 and 2, multiply Equation 1 by [tex]\( 6 \)[/tex].
- To eliminate [tex]\( y \)[/tex] from Equations 1 and 3, multiply Equation 1 by [tex]\( 1 \)[/tex].
The resulting equations are:
[tex]\[ \left\{ \begin{array}{l} x - 12y + 3z = -17 \\ x - 3z = -11 \end{array} \right. \][/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
