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Chapter 10: Problem 18

If \(x+3 y=5\) and \(3 x+y=7\), then \(x+y=\) (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Short Answer

Expert verified
Simplifying yields \( -8y + 15 = 7 \). Next, \( -8y = -8 \), so \( y = 1 \). Substituting \( y = 1 \) back, \( x = 2 \). So, \( x + y = 1 + 2 = 3 \).

Step by step solution

01

Identify the System of Equations

We start with the given equations: \( x + 3y = 5 \) and \( 3x + y = 7 \). These two equations form a system of linear equations that we need to solve.
02

Solve for One Variable

To solve these equations, let's first express one variable in terms of the other. From \( x + 3y = 5 \), solve for \( x \). Subtract \( 3y \) from both sides: \( x = 5 - 3y \).
03

Substitute the Expression into the Second Equation

Substitute \( x = 5 - 3y \) into the second equation \( 3x + y = 7 \):\[ 3(5 - 3y) + y = 7 \] Simplifying gives:\[ 15 - 9y + y = 7 \]

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Key Concepts

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

Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are characterized by variables with an exponent of one. This means that graphs of linear equations are straight lines. These equations are foundational in algebra and appear in many real-world applications.

Linear equations can have one variable, like in the form of \( ax + b = 0 \), or more than one variable, such as \( ax + by = c \). In the given system:
  • \( x + 3y = 5 \)
  • \( 3x + y = 7 \)
    • These both are linear equations with two variables, \( x \) and \( y \).
When solved together, we can find a set of values for \( x \) and \( y \) that satisfy both equations simultaneously.
Substitution Method
The substitution method is a technique for solving a system of equations by solving one of the equations for one variable in terms of the others and then substituting that expression into the other equations. This can simplify the system to make it easier to solve.

Here's how it works in our problem:
  • From \( x + 3y = 5 \), we solve for \( x \): \( x = 5 - 3y \).
  • This expression for \( x \) is then substituted into the second equation, \( 3x + y = 7 \).
  • Replacing \( x \) gives \( 3(5 - 3y) + y = 7 \).
After substitution, the problem is reduced to a single equation with one variable, simplifying our solution process.
Solving Equations
Solving equations involves finding values for the variables that make the equation true. In the context of the method used here, after substitution, we simplify and solve the resulting equation.

Start by expanding the equation:
  • \( 3(5 - 3y) + y = 7 \) simplifies to \( 15 - 9y + y = 7 \).
  • Combine like terms to get \( 15 - 8y = 7 \).
  • Subtract 15 from both sides resulting in \( -8y = -8 \).
  • Solve for \( y \) by dividing both sides by -8, giving \( y = 1 \).
With \( y = 1 \), substitute back into \( x = 5 - 3y \) to find \( x \). It leads to \( x = 5 - 3(1) = 2 \). Now, we have \( x = 2 \) and \( y = 1 \), thus, \( x + y = 3 \).
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the given exercise, algebraic techniques are used to solve a system of equations, an essential skill in algebra.

We employ various algebraic operations like:
  • Solving for a variable, as we do when converting \( x + 3y = 5 \) into \( x = 5 - 3y \).
  • Substitution, which replaces a variable with an equivalent expression.
  • Simplifying equations by combining like terms or applying arithmetic operations.
Mastering algebraic skills enables us to solve complex problems and understand relationships between variables. This forms a core part of mathematical literacy essential in advanced math and many practical scenarios.

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Most popular questions from this chapter

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) \(\frac{7}{2}\)

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) \(2(12)^7\)

If \(x+y=4 a / 5, y+z=7 a / 5\) and \(z+x=9 a / 5\), then \(x+y+z=\) (A) \(7 a / 15\) (B) \(a\) (C) \(2 a\) (D) \(3 a\) (E) \(4 a\)

If 4 percent of \((p+q)\) is 8 and \(p\) is a positive integer, what is the greatest possible value of \(q\) ? (A) 196 (B) 197 (C) 198 (D) 199 (E) 200

If \(\frac{p}{19}\) is 1 less than 3 times \(\frac{q}{19}\), then \(p\) equals which of the following expressions? (A) \(3 q+19\) (B) \(3 q+38\) (C) \(\frac{19}{2}\) (D) \(3 q-38\) (E) \(3 q-19\)
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