Given the equations, solve for the variables based on provided conditions.
Understand the Problem
The question seems to focus on solving a mathematical problem involving variables and equations related to a specific situation outlined in the content visible. It presents equations and may require simplification or substitution to find the values of the variables involved.
Answer
$$ z = 18 - d, \quad y = 4d - 54, \quad x = 4d - 54 $$
Answer for screen readers
The values of the variables expressed in terms of ( d ) are: $$ z = 18 - d $$ $$ y = 4d - 54 $$ $$ x = 4d - 54 $$
Steps to Solve
- Substituting for ( z )
From the equation ( z + d = 18 ), we can express ( z ) as: $$ z = 18 - d $$
- Simplifying the second equation
Substituting ( z ) into the second equation ( 2y - y + 4z = 18 ): $$ 2y - y + 4(18 - d) = 18 $$ This simplifies to: $$ y + 72 - 4d = 18 $$
- Rearranging the second equation
Rearranging the simplified form gives: $$ y - 4d + 72 = 18 $$ Now, simplify this equation: $$ y - 4d = 18 - 72 $$ $$ y - 4d = -54 $$
- Simplifying the third equation
Now, replace ( z ) in the third original equation ( 2x - y + 4z = 18 ): $$ 2x - y + 4(18 - d) = 18 $$ This simplifies to: $$ 2x - y + 72 - 4d = 18 $$
- Rearranging the third equation
Rearranging gives: $$ 2x - y + 72 = 18 + 4d $$ Now reformat it: $$ 2x - y = 18 + 4d - 72 $$ $$ 2x - y = 4d - 54 $$
- Solving the linear equations
Now, we have two equations to work with:
- ( y - 4d = -54 )
- ( 2x - y = 4d - 54 )
Using the first equation, express ( y ) in terms of ( d ): $$ y = 4d - 54 $$
Substituting into the second equation: $$ 2x - (4d - 54) = 4d - 54 $$
- Isolating ( x )
Solve for ( x ): $$ 2x - 4d + 54 = 4d - 54 $$ $$ 2x = 8d - 108 $$ $$ x = 4d - 54 $$
- Final values
At this point, you can express ( x ), ( y ), ( z ), and ( d ) in terms of ( d ).
The values of the variables expressed in terms of ( d ) are: $$ z = 18 - d $$ $$ y = 4d - 54 $$ $$ x = 4d - 54 $$
More Information
This set of equations represents a linear relationship among the variables ( x, y, z, ) and ( d ). Adjusting ( d ) changes the values of ( x, y, ) and ( z ).
Tips
- Not simplifying the equations correctly after substitution.
- Forgetting to rearrange and isolate terms properly.
- Confusing the variables during substitution.
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