Verified Answer :

  • Isolate the term containing \(y\) by subtracting \(7\) from both sides: \(22 - 7 = 6y\), resulting in \(15 = 6y\).
  • Solve for \(y\) by dividing both sides of the equation by \(6\): \(y = \frac{15}{6}\).
  • Simplify the fraction \(\frac{15}{6}\) to its simplest form, \(\frac{5}{2}\).
  • The final value is \(\boxed{\frac{5}{2}}\).

Explanation

  1. Analysis of the Equation

The given equation is \(22 = 6y + 7\), where \(y\) is the variable we need to solve for. This is a linear equation in one variable, representing a relationship where a constant value (22) is set equal to a linear expression involving \(y\). Our goal is to isolate \(y\) by performing algebraic operations on both sides of the equation.

  1. Isolating the Term with y

To begin isolating the variable \(y\), we first subtract the constant term \(7\) from both sides of the equation:
\[22 - 7 = 6y + 7 - 7\]
Simplifying both sides, we get:
\[15 = 6y\]
This reduces the equation to a simpler form where the variable \(y\) is multiplied by a coefficient.

  1. Solving for y

Now, we isolate \(y\) by dividing both sides of the equation by the coefficient \(6\):
\[\frac{15}{6} = \frac{6y}{6}\]
Simplifying the fractions, we have:
\[y = \frac{15}{6}\]
By dividing both the numerator and the denominator by their greatest common divisor, which is 3, we simplify the fraction:
\[y = \frac{15 \div 3}{6 \div 3} = \frac{5}{2}\]
Thus, the value of \(y\) is \(\frac{5}{2}\), or \(2.5\).

Examples

Linear equations like this are fundamental in everyday scenarios. For example, if a taxi charges a flat fee of \(\$7\) plus \(\$6\) per kilometer, and you paid \(\$22\) for a ride, you can set up the equation \(6y + 7 = 22\) to determine the distance \(y\) you traveled. Solving this equation helps you verify your travel distance based on the total cost.

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