91Ó°ÊÓ

A system of equations means having two or more equations that share the same variables. In this exercise, we have two angles whose measures are related and are thus represented by two equations.
The first equation arises from the fact that the two angles are complementary. Complementary angles add up to 90 degrees. Therefore, the sum of these two angles, denoted by variables, should equal 90.
The second equation relates the two angles further by indicating one angle's measure as a function of the other. The language of the problem provides this condition: 'The measure of the larger angle is ten more than four times the measure of the smaller angle.' Translating this from words to a mathematical equation allows us to set up our second equation.
When solving systems of equations, we often use methods like substitution or elimination to find the values of the variables. These methods simplify the equations to make finding the solution more manageable.

Understanding the properties of angles will make solving the problem easier. For complementary angles, their measures add up to 90 degrees. Knowing this helps us quickly set up one of our equations.
Another important concept is how to describe one angle's measure in terms of another. In the problem, the measure of the larger angle is defined as 'ten more than four times the measure of the smaller angle.' This allows us to create an equation where one angle's measure is directly connected to the other.
This understanding not only allows us to define our system of equations but also guides us through the problem-solving process.

One effective method for solving a system of equations is algebraic substitution. This technique involves solving one of the equations for one variable and then substituting that expression into the other equation. This way, you reduce the system from two equations to one equation with one variable.
In this case, we start with the two equations: \ x + y = 90 \ y = 4x + 10 \
We can substitute the expression found for y (\(4x + 10\)) from the second equation into the first equation: \ x + (4x + 10) = 90.
We then solve the simplified equation step by step: \ x + 4x + 10 = 90 \ 5x + 10 = 90 \ 5x = 80 \ x = 16
With the value of x known, we can put it back into the second original equation to find y: \ y = 4(16) + 10 = 64 + 10 = 74.
The algebraic substitution method is particularly powerful for solving systems of linear equations like these, making it easier to find the exact values of the variables involved.