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Chapter 10: Problem 19

If \(7 x-y=23\) and \(7 y-x=31\), then \(x+y=\) (A) 4 (B) 6 (C) 7 (D) 8 (E) 9

Short Answer

Expert verified
The sum \(x + y\) is 9.

Step by step solution

01

Understand the Given Equations

We are given two equations: \(7x - y = 23\) and \(7y - x = 31\). Our goal is to find the sum \(x + y\).
02

Set Up System of Equations

The system of equations we have is: \(7x - y = 23\) and \(7y - x = 31\). We need to solve this system to find \(x\) and \(y\).
03

Add the Two Equations

To eliminate one variable and simplify, add the equations: \( (7x - y) + (7y - x) = 23 + 31\). This simplifies to \(6x + 6y = 54\).
04

Solve for Simplified Equation

Divide the whole equation \(6x + 6y = 54\) by 6 to simplify it further, resulting in \(x + y = 9\).
05

Final Solution

The value of \(x + y\) is 9.

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Key Concepts

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

System of Equations
A system of equations is when you have two or more equations with the same set of variables. To solve these systems, you need to find the values of the variables that make all equations true at the same time. In the GMAT problem given, we have:
  • Equation 1: \(7x - y = 23\)
  • Equation 2: \(7y - x = 31\)
The goal here is to find values for \(x\) and \(y\) that satisfy both equations simultaneously.

This is crucial for solving many real-world problems where multiple conditions need to be met at once. Practice different methods such as substitution or elimination to become proficient in solving these systems.
Algebraic Manipulation
Algebraic manipulation involves changing equations in different ways to make solving for unknown variables easier. In our problem, the goal is to find \(x\) and \(y\), but directly tackling each equation doesn't help easily.

To simplify the process, we can use strategic operations:
  • Start by rearranging equations if needed or directly manipulating them to eliminate one of the variables.
  • For instance, adding Equations 1 and 2 eliminates the undesirable terms by canceling out one variable, simplifying to a single variable equation.
  • Then, dividing the equation resulting from combining the original equations helps isolate the desired sum \(x + y\).
Practicing these techniques will help you understand how different forms of equations relate and ensure quicker problem-solving in exams.
Linear Equations
Linear equations are equations of the first order, meaning they produce a straight line when graphed. They usually appear in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

In our example:
  • Equation 1: \(7x - y = 23\)
  • Equation 2: \(7y - x = 31\)
Both are linear equations. When handling linear equations, the focus is on finding where these lines intersect, which gives the solution to the system of equations.

Understanding the basics of linear equations sets you up well for solving various exercises, especially in standardized tests like the GMAT. It鈥檚 important to recognize patterns in linearity, such as consistent increases or decreases, which help simplify problem-solving processes.

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Most popular questions from this chapter

If \(3^x=81\), then \(\left(3^{x+3}\right)\left(4^{x+1}\right)=\) (A) \(5(7)^5\) (B) \(9(7)^5\) (C) \(2(12)^4\) (D) \(9(12)^5\) (E) \(2(12)^7\)

Suppose \(x=y-2=\frac{y+5}{2}\). Then \(x\) equals (A) \(\frac{1}{3}\) (B) 1 (C) \(\frac{7}{6}\) (D) 2 (E) 7

If \(x=y / 2\) and \(y=z / 2\), then \(\sqrt{x / z}=\) (A) 4 (B) 2 (C) 1 (D) \(1 / 2\) (E) \(1 / 4\)

If \(a=b / c\) and \(b=a / c\), then \(c=\) (A) \(b / a\) (B) \(a / b\) (C) \(-1\) (D) \(a\) (E) \(-b\)

Let \(P=(x+y) k\). If \(P=10\) and \(k=3\), what is the average of \(x\) and \(y\) ? (A) 0 (B) \(\frac{1}{2}\) (C) \(\frac{5}{3}\) (D) \(\frac{10}{3}\) (E) \(\frac{7}{2}\)
See all solutions

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