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Chapter 4: Problem 47

Solve by substitution. $$ \begin{aligned} &k=\frac{4}{9} h+8 \\ &-4 h+9 k=70 \end{aligned} $$

Short Answer

Expert verified
The equations result in a contradiction, hence no real solution.

Step by step solution

01

Solve the First Equation for One Variable

The first equation is given as: \[ k = \frac{4}{9}h + 8 \]Let's solve this equation for the variable \( h \).
02

Isolate the Variable

We need to isolate \( h \) on one side of the equation. First, rearrange the equation to: \[ k - 8 = \frac{4}{9}h \]Multiply both sides by 9 to get rid of the fraction: \[ 9(k - 8) = 4h \]Simplify to: \[ 9k - 72 = 4h \]Then, divide both sides by 4: \[ h = \frac{9k - 72}{4} \]
03

Substitute into the Second Equation

Now substitute \( h \) back into the second equation: \[ -4h + 9k = 70 \]Replace \( h \) with \( \frac{9k - 72}{4} \): \[ -4 \left( \frac{9k - 72}{4} \right) + 9k = 70 \]Simplify by canceling out the 4: \[ -(9k - 72) + 9k = 70 \]Which simplifies to: \[ -9k + 72 + 9k = 70 \]Combine like terms: \[ 72 = 70 \]There is no contradiction, let's continue to solve.
04

Solve for k

Notice that \( -9k + 9k \) cancels out: \[ 72 = 70 \]In this step, we identify a contradiction in the math, hence let's assume and re-evaluate our steps ensuring correctness.
05

Refine Values If Any

Reevaluate the system noting that we may have algebra accuracy checking.

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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 an algebraic technique for solving systems of equations. To use this method, you first solve one of the equations for one variable in terms of the other variable(s).
Then, you substitute this expression into the other equation(s). This helps to reduce the system to a single equation with one variable, which can be solved easily.
Substitution is particularly useful when one equation is already solved for one variable, or can be easily rearranged.
In our exercise, we begin by solving the first equation for the variable of interest.
Isolating Variables
Isolating a variable means to get the variable alone on one side of the equation.
This often involves rearranging the equation by using inverse operations such as addition, subtraction, multiplication, and division.
Algebraic Equations
Algebraic equations are mathematical statements that show the equality between two expressions.
They involve variables and constants, and can be solved using various algebraic techniques. In our example, we have two algebraic equations involving the variables h and k.
Systems of Linear Equations
Systems of linear equations consist of two or more linear equations involving the same set of variables.
The solution to the system is the set of values that satisfy all equations simultaneously.
There are various methods to solve these systems, including substitution, elimination, and graphing.
In our case, we used the substitution method to find the values of the variables that satisfy both equations.

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