Complementary angles P and Q have a sum of 90 degrees. Angle P is 6 degrees more than twice the measure of angle Q. Write a system of equations and use substitution to find the mea... Complementary angles P and Q have a sum of 90 degrees. Angle P is 6 degrees more than twice the measure of angle Q. Write a system of equations and use substitution to find the measures of angles P and Q.
Understand the Problem
The question describes two complementary angles, P and Q. It states that the measure of angle P is 6 degrees than twice the measure of angle Q. The question requires you to create a system of linear equations based on the information provided and then solve for the measures of angles P and Q using the substitution method.
Answer
$P = 62^\circ$ $Q = 28^\circ$
Answer for screen readers
$P = 62^\circ$ $Q = 28^\circ$
Steps to Solve
- Write the first equation based on the definition of complementary angles
Since angles P and Q are complementary, their sum is 90 degrees. This can be written as: $$P + Q = 90$$
- Write the second equation based on the relationship between angles P and Q
The measure of angle $P$ is 6 degrees more than twice the measure of angle $Q$. This translates to: $$P = 2Q + 6$$
- Substitute the second equation into the first equation
Substitute $P = 2Q + 6$ into $P + Q = 90$ to solve for $Q$:
$$(2Q + 6) + Q = 90$$
- Solve for Q
Simplify and solve the equation for Q:
$$3Q + 6 = 90$$
$$3Q = 90 - 6$$
$$3Q = 84$$
$$Q = \frac{84}{3}$$
$$Q = 28$$
- Solve for P
Substitute the value of $Q$ back into the equation $P = 2Q + 6$:
$$P = 2(28) + 6$$
$$P = 56 + 6$$
$$P = 62$$
$P = 62^\circ$ $Q = 28^\circ$
More Information
The sum of angles P and Q is 90 degrees, confirming they are complementary: $62 + 28 = 90$. Also, angle P (62 degrees) is indeed 6 degrees more than twice angle Q (28 degrees): $2(28) + 6 = 56 + 6 = 62$
Tips
A common mistake is misinterpreting the relationship between angles P and Q when forming the second equation. For example, writing $Q = 2P + 6$ instead of $P = 2Q + 6$. Careful reading and translating the words into mathematical expressions is crucial.
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