2x + y = 8, x - z = 2. Solve by elimination and substitution

In summary, the given equations cannot be solved by elimination or substitution due to having three variables and only two equations. However, it is possible to solve for y and z in terms of x, or to express any solution in terms of a parameter.
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Homework Statement



2x + y = 8, x - z = 2. Solve by elimination and substitution

Homework Equations



2x + y = 8, x - z = 2

The Attempt at a Solution



It cannot be solved by elimination or substitution.
 
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  • #2
You can solve for y and z in terms of x.
 
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  • #3
To add to LCKurtz's response, you have two equations with three variables. You can solve for any of the following:

  • [itex]x[/itex] and [itex]y[/itex] in terms of [itex]z[/itex]
  • [itex]x[/itex] and [itex]z[/itex] in terms of [itex]y[/itex]
  • [itex]y[/itex] and [itex]z[/itex] in terms of [itex]x[/itex]
But you cannot solve for all three.
 
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  • #4
What I would do is say that, from the first equation, y= 8- 2x, and from the second equation, z= x- 2.

That is, any (x, y, z) satisfying these equations is of the form (x, 8- 2x, x- 2).

Or if you prefer "parametric equations", (x, y, z) satisfies those equation if and only if x= t, y= 8- 2t, and z= t- 2. Those points lie on a straight line in "three-space". Geometrically, a line is determined by two points. This line is the line that passes through (0, 8, -2) and (2, 4, 0).
 
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  • #5
Oops! My "correction" was erroneous.

Apologies.
 
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  • #6
oay said:
[itex]z=x+2[/itex]

If you want to add knowledge, please don't confuse matters by getting basic things wrong.

Everything HallsofIvy said was 100% correct. Where do you think he was wrong?
 
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  • #7
Ray Vickson said:
Everything HallsofIvy said was 100% correct. Where do you think he was wrong?
Correct. Sorry to both you and HoI.
 
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Related to 2x + y = 8, x - z = 2. Solve by elimination and substitution

What is elimination and substitution in solving equations?

Elimination is a method for solving a system of equations by eliminating one variable through combining equations. Substitution is a method for solving a system of equations by substituting one equation into another to eliminate one variable.

How do you use elimination to solve 2x + y = 8 and x - z = 2?

To solve this system of equations by elimination, we need to eliminate one of the variables. We can do this by multiplying one of the equations by a constant so that when we add the equations together, one of the variables will cancel out. In this case, we can multiply the second equation by 2 and then add the equations together to eliminate the x variable.

How do you use substitution to solve 2x + y = 8 and x - z = 2?

To solve this system of equations by substitution, we need to solve one of the equations for one of the variables and then substitute that expression into the other equation. In this case, we can solve the second equation for x and then substitute that expression into the first equation. This will eliminate the x variable and allow us to solve for y.

What is the solution to the system of equations 2x + y = 8 and x - z = 2?

After using either elimination or substitution, we should end up with a single equation with one variable. Solving for that variable will give us a numerical value. In this case, the solution is x = 2, y = 4, and z = 0.

Why is it important to check the solution to a system of equations?

It is important to check the solution because there is always a possibility of making a mistake in the solving process. Checking the solution will ensure that we have found the correct values for the variables and that they satisfy all of the original equations in the system.

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