Problem 36
Question
In Exercises 33-38, justify each step of the equation. Then identify any properties of equality used to solve the equation. $$ \begin{aligned} \frac{4}{5} x &=-28 \\ \frac{5}{4}\left(\frac{4}{5} x\right) &=\frac{5}{4}(-28) \\ x &=-35 \end{aligned} $$
Step-by-Step Solution
Verified Answer
The equation \(\frac{4}{5} x = -28\) is solved by the multiplication of \(\frac{5}{4}\) to both sides, then simplifying the multiplication resulting in \(x = -35\). The properties used are: Multiplication Property of Equality to multiply \(\frac{5}{4}\) to both sides and Symmetric Property of Equality plus simplification to simplify the equation to \(x = -35\).
1Step 1: Original equation
The given equation is \(\frac{4}{5} x = -28\). This is the equation to be solved, which will involve isolating \(x\) on one side.
2Step 2: Multiplication
In the next step, \(\frac{5}{4}\) is multiplied to both sides of the equation, resulting in: \(\frac{5}{4}\left(\frac{4}{5} x\right) =\frac{5}{4}(-28)\). This step is based on the Multiplication property of equality which states that, the product of a multiply to both sides does not change the equation.
3Step 3: Simplifying
The multiplication between \(\frac{5}{4}\) and \(\frac{4}{5}\) results in 1, thus simplifying \(1x\) on the left side, while the product \(\frac{5}{4}(-28)\) on the right side simplifies to -35. Thus resulting in: \(x = -35\) . This step is based on the Symmetric property of equality and simplification.
Key Concepts
Properties of EqualityMultiplication PropertySymmetric PropertySolving Equations
Properties of Equality
When we deal with algebra equations, understanding the properties of equality is crucial. These properties are like rules that help us make valid manipulations to both sides of an equation. They ensure that the equation stays balanced, just like a scale.
- Reflexive Property: Anything is equal to itself, like a mirror. For instance, \(a = a\).
- Symmetric Property: If \(a = b\), then \(b = a\). This means you can flip the equation without changing its meaning.
- Transitive Property: If \(a = b\) and \(b = c\), then \(a = c\). If two things are each equal to a third thing, then they are equal to each other.
- Addition and Subtraction Properties: You can add or subtract the same amount from both sides without affecting the equation.
- Multiplication and Division Properties: You can multiply or divide both sides by the same non-zero number, and the equation remains true.
Multiplication Property
The Multiplication Property of Equality is a handy tool in solving equations. It states that you can multiply both sides of an equation by the same non-zero number without changing the equation's balance.
In our original exercise, we take the equation \(\frac{4}{5}x = -28\). To isolate \(x\), we need to get rid of the fraction \(\frac{4}{5}\). We multiply both sides by its reciprocal, \(\frac{5}{4}\). When you multiply \(\frac{5}{4}\) by \(\frac{4}{5}\), you end up with 1 because multiplying a number by its reciprocal equals 1.
This step transforms our equation to \(1x = \frac{5}{4}(-28)\), effectively isolating \(x\) on one side. In this step, the Multiplication Property of Equality ensures the equation remains balanced.
In our original exercise, we take the equation \(\frac{4}{5}x = -28\). To isolate \(x\), we need to get rid of the fraction \(\frac{4}{5}\). We multiply both sides by its reciprocal, \(\frac{5}{4}\). When you multiply \(\frac{5}{4}\) by \(\frac{4}{5}\), you end up with 1 because multiplying a number by its reciprocal equals 1.
This step transforms our equation to \(1x = \frac{5}{4}(-28)\), effectively isolating \(x\) on one side. In this step, the Multiplication Property of Equality ensures the equation remains balanced.
Symmetric Property
The Symmetric Property of Equality is quite intuitive. It simply means that if two things are equal, we can switch them around and they remain equal. For example, if \(x = y\), then \(y = x\).
In solving equations, especially when we simplify, this property allows us to present equations in a more conventional way, particularly emphasizing the left side as the subject. For instance, after simplifying the equation \(1x = -35\), using the symmetric property, we usually write it as \(x = -35\).
This swap doesn't change the meaning but often clears up the presentation, making it easier to recognize what variable equals what value.
In solving equations, especially when we simplify, this property allows us to present equations in a more conventional way, particularly emphasizing the left side as the subject. For instance, after simplifying the equation \(1x = -35\), using the symmetric property, we usually write it as \(x = -35\).
This swap doesn't change the meaning but often clears up the presentation, making it easier to recognize what variable equals what value.
Solving Equations
Solving equations is like unraveling a puzzle by isolating the variable. It involves systematically using properties like those discussed above. Let's break it down further:
- Identify the Goal: Find what the equation is trying to express, usually isolating a variable like \(x\).
- Apply Logical Steps: Repeat operations like addition, subtraction, multiplication, or division on both sides to simplify the equation.
- Simplify: Use arithmetic or algebraic rules to simplify each side, reducing it to a simpler form.
- Verify: Once you find a solution, plug it back into the original equation to ensure it makes both sides equal.
Other exercises in this chapter
Problem 35
$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ -(4 x) $$
View solution Problem 35
In Exercises 19-36, expand the expression as a product of factors. $$ \left[2(a-b)^{3}\right]\left[2(a-b)^{2}\right] $$
View solution Problem 36
In Exercises 23-36, write a verbal description of the algebraic expression, without using a variable. (There is more than one correct answer.) $$ x^{3}-1 $$
View solution Problem 36
$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ -(5 t) $$
View solution