Question

If \(y / x=4\) and \(x+y=5\), then \(x\) is: (A) 4 (B) 1 (C) 5 (D) \(4 / 5\) (E) \(5 / 4\)

Short answer

The value of x is 1.

Step-by-step solution

Define the equations

From the problem, we have two important equations. The first equation is given as \(\frac{y}{x} = 4\). The second equation is \(x + y = 5\).

Express y in terms of x

From the first equation \(\frac{y}{x} = 4\), we can express \(y \) in terms of \(x\). This gives us: \(y = 4x\).

Substitute y in the second equation

Substitute \(y = 4x\) into the second equation \(x + y = 5\). This becomes: \x + 4x = 5\.

Solve for x

Combine like terms: \(5x = 5\). Then solve for \(x\) by dividing both sides by 5: \(x = 1\).

Verify the solution

To ensure the solution is correct, verify by substituting \(x = 1\) back into the original equations. If \(x = 1\), then \(y = 4(1) = 4\). Check if \(1 + 4 = 5\), which is true, and \(\frac{4}{1} = 4\), which is also true.

Key concepts

Solving Equations

In algebra, solving equations involves finding the value of the unknown variable that makes the equation true. Equations are often presented with an equality sign (=), showing that two expressions are equal. The goal is to isolate the variable on one side of the equation to determine its value.

For the given problem, we have two equations: \(\frac{y}{x} = 4\) and \(x + y = 5\). Our task is to find the value of \(x\) that satisfies both equations simultaneously.

Substitution Method

The substitution method is a common technique used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This allows us to solve for one variable in terms of the other.

In this problem, we start with the first equation, \(\frac{y}{x} = 4\), and solve for \(y\) in terms of \(x\). This gives us: \(y = 4x\).

• Next, we substitute \(4x\) for \(y\) in the second equation, \(x + y = 5\).
• This results in a new equation with only one variable: \(x + 4x = 5\).
• Combining like terms, we get: \(5x = 5\).
  
  By dividing both sides of the equation by 5, we find that \(x = 1\).
  

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators (such as +, -, *, /). Unlike an equation, an expression does not have an equality sign. In the given problem, we worked with expressions when we rewrote the equations.

For instance, starting with the expression \(\frac{y}{x} = 4\) and rewriting it as \(y = 4x\) is an example of manipulating algebraic expressions.

• When we plugged \(4x\) back into the equation \(x + y = 5\), we formed the expression \(x + 4x\).
• Simplifying this to \(5x\) and solving for \(x\) required us to understand how to handle algebraic expressions properly.
  
  Remembering how to express one variable in terms of another and substitute correctly are crucial skills in algebra. These strategies help break down complex problems into manageable steps.