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Chapter 4: Problem 42

Solve by substitution. $$ \begin{gathered} k=-3 h+100 \\ 4 h+9 k=210 \end{gathered} $$

Short Answer

Expert verified
The solution is \( h = 30 \) and \( k = 10 \).

Step by step solution

01

Solve for One Variable

The first equation is given as \( k = -3h + 100 \). This equation is already solved for \( k \).
02

Substitute into the Second Equation

Substitute \( k = -3h + 100 \) into the second equation: \( 4h + 9k = 210 \). This gives:\[ 4h + 9(-3h + 100) = 210 \]
03

Simplify the Equation

Distribute the 9 in the equation:\[ 4h - 27h + 900 = 210 \]Combine like terms:\[ -23h + 900 = 210 \]
04

Solve for \( h \)

Isolate \( h \) by subtracting 900 from both sides:\[ -23h = 210 - 900 \]This simplifies to:\[ -23h = -690 \]Then divide both sides by -23:\[ h = 30 \]
05

Solve for \( k \)

Substitute \( h = 30 \) back into the first equation \( k = -3h + 100 \):\[ k = -3(30) + 100 \]This simplifies to:\[ k = -90 + 100 \]\[ k = 10 \]
06

Conclusion

The solution to the system of equations is \( h = 30 \) and \( k = 10 \).

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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 way to solve a system of equations by solving one of the equations for one variable and then substituting this solution into the other equation.
This method transforms the system into a single equation with one variable, making it easier to solve.
Let's break it down.
In the given problem, we have two equations:
  • First equation: \( k = -3h + 100 \)
  • Second equation: \( 4h + 9k = 210 \)
The first equation is already solved for \( k \), so we can directly substitute \( k \) in the second equation.
solving linear equations
Linear equations are equations that graph as straight lines.
These equations often involve one or two variables and can be written in the form \( ax + by = c \).
In our problem, we see the second equation as \( 4h + 9k = 210 \). After substituting \( k = -3h + 100 \), we get:
\( 4h + 9(-3h + 100) = 210 \)
By distributing and combining like terms, we simplify it to:
\( -23h + 900 = 210 \)
Next, we isolate and solve for \( h \).
The idea is to move all the terms involving \( h \) to one side and constants to the other side.
Subtracting 900 from both sides:
\( -23h = 210 - 900 \)
\( -23h = -690 \)
Dividing both sides by -23 gives us:
\( h = 30 \)
algebraic manipulation
Algebraic manipulation involves using basic algebra rules to simplify and solve equations.
These rules include distributive property, combining like terms, and isolating variables.
After we find \( h = 30 \), we're not done yet.
We need to find the value of \( k \).
To do this, substitute \( h = 30 \) back into the first equation \( k = -3h + 100 \):
\( k = -3(30) + 100 \)
Simplifying we get:
\( k = -90 + 100 \)
\( k = 10 \)
Hence, the system of equations is solved with \( h = 30 \) and \( k = 10 \). Using these steps ensures a clear and systematic approach to solving such problems.

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Most popular questions from this chapter

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