Understanding Supplementary Angles

Two angles are considered supplementary angles if the sum of their measures is exactly 180 degrees. This is a fundamental concept in geometry.

In this problem, we are given two angles expressed in terms of a variable 'y':

• Angle 1: \((6y + 70)^{\circ}\)
• Angle 2: \((3y + 47)^{\circ}\)

We are told that these two angles are supplementary angles. Therefore, their sum must equal 180 degrees.

Setting Up the Supplementary Angle Equation

Based on the definition of supplementary angles, we can write the following algebraic equation:

We can drop the degree symbol while solving the algebraic equation:

Solving for the Value of y

Now, we need to solve this linear equation for 'y'. The steps are as follows:

1. Combine like terms on the left side of the equation (terms with 'y' and constant terms).
2. Isolate the term with 'y'.
3. Solve for 'y'.

Let's combine the 'y' terms and the constant terms:

Now, subtract 117 from both sides of the equation to isolate the term with 'y':

Finally, divide both sides by 9 to find the value of 'y':

Verifying the Solution

To verify our answer, we can substitute \(y=7\) back into the expressions for the two angles:

• Angle 1: \((6 \times 7 + 70)^{\circ} = (42 + 70)^{\circ} = 112^{\circ}\)
• Angle 2: \((3 \times 7 + 47)^{\circ} = (21 + 47)^{\circ} = 68^{\circ}\)

Now, let's check if their sum is 180 degrees:

Since the sum is 180 degrees, our value of \(y=7\) is correct for the supplementary angles.

The value of \(y\) is 7.

Matching the Answer with Options

Let's compare our result with the given options:

• A. 12
• B. 15
• C. 7
• D. 10

Our calculated value for \(y\) is 7, which matches option C.

Revision Table: Key Concepts in Angles

Additional Information: Solving Linear Equations

The problem required solving a basic linear equation. Here is a quick recap of the steps involved in solving an equation of the form \(ax + b = c\):

1. Combine Like Terms: If there are multiple terms with the variable or multiple constant terms on one side, combine them first. (In our problem, we combined \(6y\) and \(3y\), and \(70\) and \(47\)).
2. Isolate the Variable Term: Use addition or subtraction to move the constant term (\(b\)) to the other side of the equation. Whatever operation you perform on one side, you must perform the same operation on the other side to keep the equation balanced. \(ax = c - b\). (In our problem, we subtracted 117 from both sides).
3. Solve for the Variable: Use multiplication or division to isolate the variable (\(x\)). If the variable is multiplied by a coefficient (\(a\)), divide both sides by that coefficient. \(x = \frac{c - b}{a}\). (In our problem, we divided both sides by 9).

Solving these equations is a critical skill for many geometry problems.