Question

Algebra If \(m \angle 1=94, m \angle 2=16-5 x\), and \(m \angle 1=m \angle 2+10\), find the value of \(x\).

Short answer

\( x = -13.6 \)

Step-by-step solution

Understand the Problem

We are given that the measures of angles are related by the equation \( m \angle 1 = m \angle 2 + 10 \). We need to find the value of \( x \) using the values given: \( m \angle 1 = 94 \) and \( m \angle 2 = 16 - 5x \).

Substitute Known Values

Substitute the known values into the equation \( m \angle 1 = m \angle 2 + 10 \). We know \( m \angle 1 = 94 \) and \( m \angle 2 = 16 - 5x \), so we can write the equation as \( 94 = (16 - 5x) + 10 \).

Simplify the Equation

Simplify the equation obtained in the previous step by combining like terms.\[ 94 = 16 + 10 - 5x \] This simplifies to \[ 94 = 26 - 5x \].

Solve for \( x \)

Rearrange the equation to solve for \( x \). Start by subtracting 26 from both sides to isolate the term with \( x \) on one side: \[ 94 - 26 = -5x \]. This simplifies to \[ 68 = -5x \].

Calculate \( x \)

Finally, solve for \( x \) by dividing both sides of the equation by -5. \[ x = \frac{68}{-5} \], resulting in \[ x = -13.6 \].

Key concepts

Angle Relationships

Understanding angle relationships is crucial when solving problems involving different angle measures. In this exercise, we deal with two angles, Angle 1 and Angle 2, that are related by an equation. We know that Angle 1 is 10 degrees more than Angle 2. This means Angle 1 is larger than Angle 2. This relationship is expressed as:

• Angle 1 = Angle 2 + 10

When given algebraic expressions for these angles, we can use this relationship to set up an equation. This equation allows us to find unknown variables such as \( x \), which is part of the expression for Angle 2. Understanding these relationships helps us correctly formulate and solve the problem.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. In our exercise, the expression for Angle 2 is given as \( 16 - 5x \). This is an algebraic expression where \( x \) is the variable we need to solve for. Expressions like these are key components in setting up equations. They are used to represent quantitative relationships between various elements. To work with them effectively, you need to:

• Identify the terms and how they interact
• Understand which operations are performed and the sequence of those operations

Handling algebraic expressions correctly is essential to solving for unknowns.

Equation Simplification

Simplifying equations is a vital step in solving them. It means breaking down complex equations into simpler forms that are easier to work with. Steps to simplify might include combining like terms or rearranging terms for clarity.In this exercise, to simplify, we first substituted known values:

• Substitute the given value of Angle 1 and the expression for Angle 2 into the equation.

Then, we need to combine like terms:

• Combine numbers on one side of the equation to simplify it further.

This simplification leads us closer to finding our unknown variable \( x \). Lastly, rearrange and solve the equation to isolate the unknown. This technique helps us understand how to manipulate equations to solve problems with ease.