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Chapter 4: Problem 63

(a) solve one of the equations for a variable and solve by substitution. (b) if there is one solution, check. $$ \begin{aligned} &7 x+y=8 \\ &7 x+y=13 \end{aligned} $$

Short Answer

Expert verified
No solution, as the system is inconsistent.

Step by step solution

01

Solve one of the equations for y

Solve the first equation for y. We start with the equation:\[ 7x + y = 8 \]Subtract 7x from both sides to isolate y:\[ y = 8 - 7x \]
02

Substitute y in the second equation

Now substitute \( y = 8 - 7x \) into the second equation:\[ 7x + y = 13 \]This gives:\[ 7x + (8 - 7x) = 13 \]
03

Simplify and solve for x

Simplify the equation:\[ 7x + 8 - 7x = 13 \]Combine like terms:\[ 8 = 13 \]Since this statement is false, there is no solution. The system of equations is inconsistent.

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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 systems of equations by isolating one variable in one of the equations and then substituting that expression into the other equation. This allows us to solve for the variables in a step-by-step manner. In the given problem, we isolated \( y \) in the first equation. Starting from \( 7x + y = 8 \), we subtracted \( 7x \) from both sides, giving us \( y = 8 - 7x \). Once we had \( y \) in terms of \( x \), we substituted this expression into the second equation, \( 7x + y = 13 \). This gave us a new equation in terms of \( x \) only, which we then simplified. This method helps you find the value of one variable, which can then be used to find the other variable.
inconsistent systems
An inconsistent system of equations is one that has no solution. This means the equations represent parallel lines that never intersect. In our example, after substituting \( y = 8 - 7x \) into the second equation, we simplified it to \( 8 = 13 \). Since this statement is mathematically impossible (8 does not equal 13), it indicates that there's no solution to the system. Thus, the given equations are inconsistent. When dealing with systems of equations, always look for such contradictions. They help in identifying whether the system is consistent (having at least one solution) or inconsistent (no solution at all).
algebraic manipulation
Algebraic manipulation is a key skill needed to solve systems of equations. This involves operations like addition, subtraction, multiplication, and division to rearrange and simplify equations. In our exercise, we performed algebraic manipulation by:
  • Subtracting \( 7x \) from both sides of the first equation to isolate \( y \).
  • Substituting \( y \) into the second equation to eliminate one variable.
  • Simplifying the resulting equation to check for solutions or inconsistencies.
Each step involves careful calculation to maintain the equality of the equation. Practicing these manipulations will make you more proficient in solving a variety of algebraic problems. Always double-check your work to ensure accuracy, especially in multi-step problems.

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Most popular questions from this chapter

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