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Magazine Chapter 8: Problem 83
Solve the system of equations \(x+y-z=7,\) \(3 x+y-2 z=12,\) and \(x+2 y+z=44\).
Short Answer
Expert verified
The solution is \(x = 9, y = 11, z = 13\).
Step by step solution
01
- Write down the system of equations
The given system of equations is:1) \(x + y - z = 7\)2) \(3x + y - 2z = 12\)3) \(x + 2y + z = 44\)
02
- Eliminate one variable
First, eliminate \(z\) from two of the equations. Multiply equation 1 by 2 and subtract from equation 2:\(2(x + y - z) = 2 \cdot 7 = 14\)This gives \(2x + 2y - 2z = 14\)
03
- Subtract the modified equation
Now, subtract this from equation 2:\( \(3x + y - 2z = 12\) - \(2x + 2y - 2z = 14\) \)This simplifies to:\( x - y = -2\) -- Equation (4)
04
- Eliminate \(z\) again using different pairs
Add equations 1 and 3:\( (x + y - z) + (x + 2y + z) = 7 + 44 \)This gives \(2x + 3y = 51\) -- Equation (5)
05
- Solve the reduced system
Now solve equations (4) and (5):\ \(x - y = -2\)\ \(2x + 3y = 51\)Substitute \(x = y - 2\) into equation (5): \2(y-2) + 3y = 51\2y - 4 + 3y = 51\Simplifies to:\5y - 4 = 51\, so \5y = 55\, and \y = 11\
06
- Find \(x\)
Substitute \(y = 11\) back into \(x = y - 2\):\x = 11 - 2\This gives \(x = 9\)
07
- Find \(z\)
Substitute \(x = 9\) and \(y = 11\) into the original first equation:\ 9 + 11 - z = 7\This simplifies to:\ 20 - z = 7\So \z = 13\
08
- Verify the solution
Substitute \(x = 9, y = 11, z = 13\) into the original equations:1) \(9 + 11 - 13 = 7\)2) \(3(9) + 11 - 2(13) = 12\)3) \(9 + 2(11) + 13 = 44\)All equations hold true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
linear equations
A linear equation is an equation that describes a straight line when graphed. Each term in a linear equation is either a constant or the product of a constant and a single variable. Linear equations have variables that are only to the first power and have no products of variables. For example, in the system given: 1) \(x + y - z = 7\) 2) \(3x + y - 2z = 12\) 3) \(x + 2y + z = 44\) All the equations are linear. To solve these, we often look for the values of \(x\), \(y\), and \(z\) that satisfy all the equations at the same time.
substitution method
The substitution method is a way to solve a system of equations where you solve one equation for one variable and then substitute that solution into the other equations. Here鈥檚 a quick guide to using substitution in our problem: 1. Take one of the simpler equations and solve for one variable. For instance, from equation (4) \(x - y = -2\), we solve for \(x\): \(x = y - 2\) 2. Substitute \(x = y - 2\) into equation (5): \(2x + 3y = 51\) becomes \(2(y - 2) + 3y = 51\). 3. Solve the new equation for \(y\). 4. Substitute back to find \(x\), and finally substitute these back into one of the original equations to find \(z\).
elimination method
The elimination method involves adding or subtracting equations to eliminate one of the variables. Follow these steps to use elimination: 1. To eliminate \(z\), multiply equation (1) by 2: \(2(x + y - z) = 14\), so \(2x + 2y - 2z = 14\). 2. Subtract this result from equation (2): \((3x + y - 2z) - (2x + 2y - 2z) = 12 - 14\). This simplifies to: \(x - y = -2\). 3. To eliminate \(z\) again, add equations (1) and (3): \((x + y - z) + (x + 2y + z) = 7 + 44\). This results in: \(2x + 3y = 51\). Now, you have two equations with two variables: \(x - y = -2\) and \(2x + 3y = 51\). 4. Solve these reduced equations to find \(x\), \(y\), and subsequently \(z\).
verifying solutions
Verifying solutions is the final and crucial step to ensure that the found values indeed satisfy all given equations. In our example: 1. After solving for \(x = 9\), \(y = 11\), and \(z = 13\), substitute these values back into the original equations: 2. For \(x + y - z = 7\): \(9 + 11 - 13 = 7\). This holds true. 3. For \(3x + y - 2z = 12\): \(3(9) + 11 - 2(13) = 12\). This also holds true. 4. For \(x + 2y + z= 44\): \(9 + 2(11) + 13 = 44\). This equally holds true. When all substituted values satisfy the original equations, the solution is verified.
