Problem 66 - Intermediate Algebra

Question

Driving Big Rigs. The two angle measures shown in the illustration of the jack-knifed trailer can be found by solving the system: $$ \left\\{\begin{array}{l} x+y=180 \\ y=4 x+5 \end{array}\right. $$ Find \(x\) and \(y\) (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The values are \(x = 35\) and \(y = 145\).

Step 1: Setup the System of Equations

Identify the given system of equations from the problem:1. Equation 1: \(x + y = 180\)2. Equation 2: \(y = 4x + 5\).

Step 2: Substitute Equation

Substitute the value of \(y\) from Equation 2 into Equation 1 to find the value of \(x\). So, the equation becomes:\[x + (4x + 5) = 180\].

Step 3: Simplify and Solve for x

Combine like terms and solve for \(x\):\(5x + 5 = 180\).Subtract 5 from both sides:\(5x = 175\).Divide both sides by 5:\(x = 35\).

Step 4: Substitute Back to Find y

Now that we have \(x = 35\), substitute it back into Equation 2 to find \(y\):\[y = 4(35) + 5\].

Step 5: Calculate y

Perform the arithmetic to find the value of \(y\):\[y = 140 + 5 = 145\].

Step 6: Verify the Solution

Check both equations with \(x = 35\) and \(y = 145\) to ensure they are satisfied:1. For \(x + y = 180\):\(35 + 145 = 180\), which is correct.2. For \(y = 4x + 5\):\(145 = 4(35) + 5\), which is also correct.

Key Concepts

Understanding Angle Measures in Geometry

Angles are fundamental parts of geometry. They represent the space between two intersecting lines, expressed in degrees. In any geometric setup, like the jack-knifed trailer, angle measures can provide valuable insights.

In this particular problem, the two angles are represented by variables \(x\) and \(y\). We see that together, these angles form a linear pair, which means their sum is 180 degrees. This is because a straight line create a 180-degree angle. In the given system of equations, this is represented as \(x + y = 180\).

Understanding that angle measures interact in such a way allows us to set up equations to solve unknown angles. Recognizing these relationships is key when dealing with geometry problems.

Applying the Substitution Method Effectively

The substitution method is a powerful algebraic tool used to solve systems of equations. It's particularly useful when one of the equations already has a variable isolated, as seen with \(y = 4x + 5\) in our problem.

Here's how the substitution works step-by-step:

• Since the second equation expresses \(y\) in terms of \(x\), substitute \(4x + 5\) for \(y\) in the first equation.
• This transforms our first equation from \(x + y = 180\) to \(x + (4x + 5) = 180\).
• This new equation only has one variable, \(x\), making it much simpler to solve.

Once \(x\) is found, its value is substituted back into the original equation to solve for \(y\). This method not only makes solving systems efficient but also reinforces understanding of algebraic manipulation.

Exploring Linear Equations in Systems

Linear equations are equations of the first degree, meaning they produce straight lines when graphed. They are essential in algebra for modeling relationships between variables.

In the given system, both equations are linear:

• \(x + y = 180\) represents a relationship where the sum of two variables is a constant.
• \(y = 4x + 5\) is another linear equation showing a direct relationship between \(x\) and \(y\).

Linear equations like these allow you to find specific values of \(x\) and \(y\) through solving methods such as substitution.

Moreover, these equations help understand geometry in the context of real-world problems, like measuring angles in our jack-knifed trailer example. The consistency they offer is what makes them so widely applicable in various fields like physics, engineering, and economics.