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Chapter 5: Problem 36

Solve the system using any method. $$ \begin{array}{l} 2 x-7 y=2400 \\ -4 x+1800=y \end{array} $$

Short Answer

Expert verified
The solution is \(x = 500\) and \(y = -200\).

Step by step solution

01

- Write down the equations

The system of equations is: 1. \(2x - 7y = 2400\) 2. \(-4x + 1800 = y\)
02

- Substitute Equation 2 into Equation 1

Use the second equation \(y = -4x + 1800\) and substitute this expression for y in the first equation: \[2x - 7(-4x + 1800) = 2400\]
03

- Simplify the equation

Distribute and combine like terms: \[2x + 28x - 12600 = 2400\] \[30x - 12600 = 2400\]
04

- Solve for x

Isolate \(x\) by adding 12600 to both sides and then dividing by 30: \[30x = 15000\] \[x = 500\]
05

- Solve for y

Substitute \(x = 500\) back into the second equation to solve for y: \[y = -4(500) + 1800\] \[y = -2000 + 1800\] \[y = -200\]
06

- Solution verification

Verify the solution by substituting \(x = 500\) and \(y = -200\) into the original equations to ensure both are satisfied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method
The substitution method is a key technique for solving systems of linear equations. In this method, one of the equations is solved for one variable in terms of the other variable. This expression is then substituted into the other equation. This helps in reducing the system to a single equation with one variable.
In our example, we have two equations:
1. \(2x - 7y = 2400\)
2. \(-4x + 1800 = y\)
We solve the second equation for y: \(y = -4x + 1800\).
Next, we substitute this expression into the first equation to solve for x.
Benefits of the substitution method:
  • It simplifies the process by reducing the number of variables.
  • It is straightforward and systematic.
  • It is helpful when one equation is already solved for a variable.
linear equations
Linear equations form the backbone of algebra. They are equations that make 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.
In our problem, we have two linear equations:
\(2x - 7y = 2400\)
\(-4x + 1800 = y\)
These equations describe relationships between the variables x and y. They are called 'linear' because their graph will be a straight line.
Understanding the structure of linear equations:
  • They typically have constants and variables.
  • Each term is of the first degree (no exponents other than 1).
  • They can have multiple variables but with no terms that are products of variables.
algebraic manipulation
Algebraic manipulation involves rearranging equations and expressions to isolate a variable or simplify the equation. This is frequently used in solving systems of equations.
In our step-by-step solution, we performed various algebraic manipulation techniques:
Step 3: Simplified the substituted equation \(2x - 7(-4x + 1800) = 2400\) to \(2x + 28x - 12600 = 2400\).
Step 4: Solved for x:
\(30x - 12600 = 2400\)
Added 12600 to both sides:
\(30x = 15000\)
Finally, divided by 30 to find:
\(x = 500\)
The essence of algebraic manipulation is dealing with equations through:
  • Combining like terms.
  • Distributing to remove parentheses.
  • Addition or subtraction of the same quantity to both sides.
  • Division or multiplication by a nonzero number to isolate the variable.
solution verification
Verification confirms that our solution satisfies the original equations. This step ensures the accuracy of our solution.
In our example, after solving for \(x = 500\) and \(y = -200\), we plug these values back into the original equations:
First equation: \(2(500) - 7(-200) = 2400\)
Simplifies to:
\(1000 + 1400 = 2400\)
which is true.
Second equation: \(-4(500) + 1800 = -200\)
Simplifies to:
\(-2000 + 1800 = -200\)
which is true.
Verification steps:
  • Substitute the found values into each original equation.
  • Confirm the left-hand side equals the right-hand side for both equations.
  • If both equations hold true, the solution is correct.

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