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Chapter 4: Problem 46

Solve by substitution. $$ \begin{aligned} &-8 c+5 d=7 \\ &d=17 c-14 \end{aligned} $$

Short Answer

Expert verified
c = 1, d = 3

Step by step solution

01

- Substitute the expression for d

Substitute the given expression for d ( d = 17c - 14 ) into the first equation (-8c + 5d = 7 ).
02

- Expand and simplify

Replace 'd' in the first equation with '17c - 14': -8c + 5(17c - 14) = 7 . Then, expand and simplify the equation: -8c + 85c - 70 = 7 .
03

- Combine like terms

Combine the terms involving 'c': 77c - 70 = 7 .
04

- Solve for c

Isolate 'c' by adding 70 to both sides: 77c = 77 . Then divide both sides by 77: c = 1 .
05

- Solve for d

Use the value of 'c' to solve for 'd' using d = 17c - 14 : d = 17(1) - 14 , which simplifies to d = 3 .

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Key Concepts

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

solving linear equations
Linear equations involve expressions that set polynomials equal to each other. They are of the form: \[ax + by = c\] where 'a', 'b', and 'c' are constants, and 'x' and 'y' are variables. To solve these equations, we usually aim to isolate one variable. This makes it easier to understand what value makes the equation true.
In this example, we have a system of linear equations:
\[-8c + 5d = 7 \]
\[d = 17c - 14 \]
The steps to solve these equations involve substitution, simplifying expressions, and solving for one variable. Once you have one variable, it becomes straightforward to find the value of the other.
substitution method
The substitution method is a technique used to solve systems of equations. It involves isolating one variable in one of the equations and then substituting this expression into the other equation.
Let's look at how it works in our example:
Given:
\[-8c + 5d = 7 \]
\[d = 17c - 14 \]
First, we substitute the second equation into the first. This means wherever 'd' is in the first equation, we'll replace it with '17c - 14'. The aim is to have only one variable in the equation making it easier to solve.
This simplifies our problem to:
\[-8c + 5(17c - 14) = 7 \].
system of equations
A system of equations consists of two or more equations with multiple variables. The goal is to find values for these variables that satisfy all the equations in the system.
In our case, we have two linear equations:
\[-8c + 5d = 7 \]
\[d = 17c - 14 \]
We solve these systems using methods like substitution or elimination. The essence of solving these systems is to find common solution sets. This involves algebraic manipulation to reduce the two equations into simpler forms and then solving them step by step.
algebraic manipulation
Algebraic manipulation is the process of rewriting equations using various algebraic properties to isolate and solve for variables. It involves operations like addition, subtraction, multiplication, division, and combining like terms.
For our problem,
1. Substitute:\[d = 17c - 14 \] into \[-8c + 5d = 7 \].
2. Expand: \[-8c + 5(17c - 14) = 7 \].
3. Simplify: \[-8c + 85c - 70 = 7 \].
4. Combine like terms: \[77c - 70 = 7 \].
5. Solve for 'c': \[77c = 77 \]
\[c = 1 \].
Finally, substitute 'c' back to find 'd': \[d = 17(1) - 14 \]
\[d = 3 \].
This step-by-step manipulation simplifies complex equations into easy-to-solve forms.

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