Identify the mathematical property used to justify this conclusion and state its symbolic definition using variables a, b, and c.

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The **Subtraction Property of Equality** states that when the same quantity is subtracted from both sides of an equation, the two sides remain equal. Mathematically, for any numbers $$a$$, $$b$$, and $$c$$:

If $$a = b$$,
then $$a - c = b - c$$

This property is a fundamental mathematical tool for solving equations. By applying it, one can systematically remove unwanted terms from one side of an equation by subtracting them from both sides, which helps to isolate the unknown variable by itself.

Subtraction Property of Equality

To solve the equation $$x + 19 = -27$$, the objective is to isolate $$x$$ by reversing the addition of $$19$$. Apply the Subtraction Property of Equality by subtracting $$19$$ from both sides:

$$x + 19 - 19 = -27 - 19$$

The left side simplifies to $$x$$ because $$19 - 19 = 0$$. The right side becomes $$-27 - 19 = -46$$. Therefore, $$x = -46$$.

To check, substitute $$-46$$ into the original equation: $$-46 + 19 = -27$$, which simplifies to $$-27 = -27$$. This true statement confirms that $$x = -46$$ is the correct solution.

Solving $$x + 19 = -27$$ Using the Subtraction Property

To solve the equation $$y + 37 = -13$$, the goal is to isolate the variable $$y$$ on one side. Since $$37$$ is being added to $$y$$, the addition can be undone by applying the Subtraction Property of Equality — subtract $$37$$ from both sides:

$$y + 37 - 37 = -13 - 37$$

On the left side, $$37 - 37 = 0$$, leaving just $$y$$. On the right side, $$-13 - 37 = -50$$. This simplifies to $$y = -50$$.

To verify the solution, substitute $$-50$$ back into the original equation: $$-50 + 37 = -13$$, which simplifies to $$-13 = -13$$. Since this is a true statement, $$y = -50$$ is confirmed as the correct solution.

Solving $$y + 37 = -13$$ Using the Subtraction Property

To translate and solve the sentence "Eleven more than $$x$$ is equal to 54," apply the systematic translation process:

1. **Locate the "equals" word:** The phrase "is equal to" translates to an equals sign ($$=$$).
2. **Translate the left side:** "Eleven more than $$x$$" indicates that $$11$$ is added to $$x$$, translating to the expression $$x + 11$$.
3. **Translate the right side:** The right side is simply $$54$$.

Combining these elements results in the equation $$x + 11 = 54$$.

To solve the equation, isolate the variable by subtracting $$11$$ from both sides:
$$x + 11 - 11 = 54 - 11$$
Simplifying this yields the solution:
$$x = 43$$

Finally, check the answer by substituting $$43$$ back into the original sentence: "Is 54 eleven more than 43?"
$$43 + 11 \stackrel{?}{=} 54$$
$$54 = 54 \checkmark$$

Translating and Solving 'Eleven more than $$x$$ is equal to 54'

After formulating an algebraic equation from a word problem, the subsequent phase is to find the value of the unknown variable using standard algebraic techniques. For example, given the equation $$w + 28 = 57$$, where $$w$$ represents the missing weight from the 1st month in a real-world scenario, the variable can be isolated by subtracting $$28$$ from both sides. Evaluating $$w + 28 - 28 = 57 - 28$$ yields $$w = 29$$, which successfully identifies the numerical value required to solve the problem.

Example of Solving an Equation in a Word Problem

To translate and solve the sentence "The sum of three-eighths and $$x$$ is one-half," apply the systematic sentence-to-equation translation process:

1. **Locate the "equals" word:** The word "is" translates to an equals sign ($$=$$).
2. **Translate the left side:** "The sum of three-eighths and $$x$$" uses the keyword *sum* to signal addition between $$\frac{3}{8}$$ and $$x$$, giving $$\frac{3}{8} + x$$.
3. **Translate the right side:** "One-half" is simply $$\frac{1}{2}$$.

Combining these elements yields the equation $$\frac{3}{8} + x = \frac{1}{2}$$.

To solve, isolate the variable by subtracting $$\frac{3}{8}$$ from both sides (Subtraction Property of Equality):

$$\frac{3}{8} + x - \frac{3}{8} = \frac{1}{2} - \frac{3}{8}$$

On the left side, $$\frac{3}{8} - \frac{3}{8} = 0$$, leaving just $$x$$. On the right side, the fractions $$\frac{1}{2}$$ and $$\frac{3}{8}$$ have different denominators, so rewrite $$\frac{1}{2}$$ with the common denominator $$8$$: $$\frac{1}{2} = \frac{4}{8}$$. Then subtract:

$$x = \frac{4}{8} - \frac{3}{8} = \frac{1}{8}$$

Finally, verify by substituting $$\frac{1}{8}$$ for $$x$$ in the original equation:

$$\frac{3}{8} + \frac{1}{8} \stackrel{?}{=} \frac{1}{2}$$

$$\frac{4}{8} \stackrel{?}{=} \frac{1}{2}$$

$$\frac{1}{2} = \frac{1}{2} \checkmark$$

Because both sides are equal, $$x = \frac{1}{8}$$ is confirmed as the correct solution. This example illustrates how the keyword "sum" in a sentence signals an addition equation involving fractions, which is solved by subtracting the known fraction from both sides. When that subtraction involves fractions with unlike denominators, a common denominator must be found before the numerators can be combined.

Translating and Solving 'The sum of three-eighths and $$x$$ is one-half'

To solve $$7x + 8 = -13$$, begin by identifying which side of the equation holds the variable term. Because $$7x$$ appears only on the left, designate the left as the **variable side** and the right as the **constant side**.

**Step 1 — Remove the misplaced constant from the variable side:** The constant $$8$$ is on the variable side, so subtract $$8$$ from both sides using the Subtraction Property of Equality:

$$7x + 8 - 8 = -13 - 8$$

$$7x = -21$$

Now all variable terms are on the left and only a constant remains on the right.

**Step 2 — Isolate the variable using the Division Property of Equality:** Because $$x$$ is multiplied by the coefficient $$7$$, divide both sides by $$7$$:

$$\frac{7x}{7} = \frac{-21}{7}$$

$$x = -3$$

**Step 3 — Check by substitution:** Replace $$x$$ with $$-3$$ in the original equation:

$$7(-3) + 8 \stackrel{?}{=} -13$$

$$-21 + 8 \stackrel{?}{=} -13$$

$$-13 = -13 \checkmark$$

Because both sides are equal, $$x = -3$$ is confirmed as the correct solution. This example shows that when the variable appears on only one side of the equation, any constants on that same side must first be moved to the opposite side before the coefficient can be divided away.

Solving $$7x + 8 = -13$$ by Collecting Constants

To solve $$9x = 8x - 6$$, first observe that variable terms appear on **both** sides of the equation, while the constant $$-6$$ appears only on the right. Designate the left side as the **variable side** and the right side as the **constant side**.

**Step 1 — Remove the variable term from the constant side:** Since $$8x$$ is on the constant side, subtract $$8x$$ from both sides using the Subtraction Property of Equality:

$$9x - 8x = 8x - 8x - 6$$

**Step 2 — Simplify:** On the left, $$9x - 8x = x$$. On the right, $$8x - 8x = 0$$, leaving just $$-6$$:

$$x = -6$$

Because the coefficient of $$x$$ is already $$1$$, no further division is needed.

**Step 3 — Check by substitution:** Replace $$x$$ with $$-6$$ in the original equation:

$$9(-6) \stackrel{?}{=} 8(-6) - 6$$

$$-54 \stackrel{?}{=} -48 - 6$$

$$-54 = -54 \checkmark$$

Because both sides are equal, $$x = -6$$ is confirmed as the correct solution. This example demonstrates that when the variable appears on both sides of an equation, the first step is to subtract the smaller variable term from both sides so that all variable terms are collected onto one side.

Solving $$9x = 8x - 6$$ by Collecting Variables

To solve $$7x + 5 = 6x + 2$$, observe that variable terms ($$7x$$ and $$6x$$) and constant terms ($$5$$ and $$2$$) appear on **both** sides of the equation. Because $$7$$ is greater than $$6$$, designate the left side as the **variable side** and the right side as the **constant side**.

**Step 1 — Remove the variable term from the constant side:** Since $$6x$$ is on the constant side, subtract $$6x$$ from both sides using the Subtraction Property of Equality:

$$7x - 6x + 5 = 6x - 6x + 2$$

Combine like terms: $$x + 5 = 2$$. Now the variable appears only on the left.

**Step 2 — Remove the constant from the variable side:** The constant $$5$$ is on the variable side, so subtract $$5$$ from both sides:

$$x + 5 - 5 = 2 - 5$$

$$x = -3$$

Because the coefficient of $$x$$ is already $$1$$, no further division is needed.

**Step 3 — Check by substitution:** Replace $$x$$ with $$-3$$ in the original equation:

$$7(-3) + 5 \stackrel{?}{=} 6(-3) + 2$$

$$-21 + 5 \stackrel{?}{=} -18 + 2$$

$$-16 = -16 \checkmark$$

Because both sides are equal, $$x = -3$$ is confirmed as the correct solution. Unlike equations where only variables or only constants need to be collected, this example required **two** collecting steps — first gathering all variable terms onto the left side and then gathering all constant terms onto the right side — before the solution could be read directly.

Solving $$7x + 5 = 6x + 2$$ by Collecting Variables and Constants

To solve $$7a - 3 = 13a + 7$$, observe that variable terms ($$7a$$ and $$13a$$) and constant terms ($$-3$$ and $$7$$) appear on **both** sides of the equation. Because $$13 > 7$$, designate the right side as the **variable side** and the left side as the **constant side**.

**Step 1 — Remove the variable term from the constant side:** Since $$7a$$ is on the constant side, subtract $$7a$$ from both sides using the Subtraction Property of Equality:

$$7a - 7a - 3 = 13a - 7a + 7$$

Combine like terms: $$-3 = 6a + 7$$. Now the variable appears only on the right.

**Step 2 — Remove the constant from the variable side:** The constant $$7$$ is on the variable side, so subtract $$7$$ from both sides:

$$-3 - 7 = 6a + 7 - 7$$

$$-10 = 6a$$

**Step 3 — Isolate the variable using the Division Property of Equality:** Because $$a$$ is multiplied by the coefficient $$6$$, divide both sides by $$6$$:

$$\frac{-10}{6} = \frac{6a}{6}$$

Simplify the fraction by dividing numerator and denominator by their greatest common factor $$2$$:

$$-\frac{5}{3} = a$$

**Step 4 — Check by substitution:** Replace $$a$$ with $$-\frac{5}{3}$$ in the original equation:

$$7\left(-\frac{5}{3}\right) - 3 \stackrel{?}{=} 13\left(-\frac{5}{3}\right) + 7$$

$$-\frac{35}{3} - \frac{9}{3} \stackrel{?}{=} -\frac{65}{3} + \frac{21}{3}$$

$$-\frac{44}{3} = -\frac{44}{3} \checkmark$$

Because both sides are equal, $$a = -\frac{5}{3}$$ is confirmed as the correct solution. This example differs from earlier ones in two important ways: (1) the **right** side is chosen as the variable side because its coefficient ($$13$$) is larger, demonstrating that either side can serve as the variable side; and (2) the final division does not produce a whole number, so the answer must be expressed as a **simplified fraction**.

Solving $$7a - 3 = 13a + 7$$ by Collecting Variables and Constants

To solve $$\frac{5}{4}x + 6 = \frac{1}{4}x - 2$$, observe that both sides contain a variable term with a fractional coefficient as well as a constant term. Because $$\frac{5}{4} > \frac{1}{4}$$, designate the left side as the **variable side** and the right side as the **constant side**.

**Step 1 — Remove the variable term from the constant side:** Since $$\frac{1}{4}x$$ is on the constant side, subtract $$\frac{1}{4}x$$ from both sides using the Subtraction Property of Equality:

$$\frac{5}{4}x - \frac{1}{4}x + 6 = \frac{1}{4}x - \frac{1}{4}x - 2$$

Combine the like terms on the left: the fractional coefficients share the denominator $$4$$, so $$\frac{5}{4} - \frac{1}{4} = \frac{4}{4} = 1$$. The equation simplifies to $$x + 6 = -2$$. Now the variable appears only on the left.

**Step 2 — Remove the constant from the variable side:** The constant $$6$$ is on the variable side, so subtract $$6$$ from both sides:

$$x + 6 - 6 = -2 - 6$$

$$x = -8$$

Because the coefficient of $$x$$ is already $$1$$, no further division is needed.

**Step 3 — Check by substitution:** Replace $$x$$ with $$-8$$ in the original equation:

$$\frac{5}{4}(-8) + 6 \stackrel{?}{=} \frac{1}{4}(-8) - 2$$

$$-10 + 6 \stackrel{?}{=} -2 - 2$$

$$-4 = -4 \checkmark$$

Because both sides are equal, $$x = -8$$ is confirmed as the correct solution. This example demonstrates that the collecting technique works identically when the variable terms have **fractional coefficients**. When two fractions with the same denominator are subtracted — here $$\frac{5}{4}x - \frac{1}{4}x$$ — the numerators are combined over the common denominator, and the result may simplify to a whole number, making the remaining steps straightforward.

Solving $$\frac{5}{4}x + 6 = \frac{1}{4}x - 2$$ by Collecting Variables and Constants

To solve $$7.8x + 4 = 5.4x - 8$$, observe that variable terms ($$7.8x$$ and $$5.4x$$) and constant terms ($$4$$ and $$-8$$) appear on **both** sides of the equation. Because $$7.8 > 5.4$$, designate the left side as the **variable side** and the right side as the **constant side**.

**Step 1 — Remove the variable term from the constant side:** Since $$5.4x$$ is on the constant side, subtract $$5.4x$$ from both sides using the Subtraction Property of Equality:

$$7.8x - 5.4x + 4 = 5.4x - 5.4x - 8$$

Combine like terms on the left by subtracting the decimal coefficients: $$7.8 - 5.4 = 2.4$$. The equation simplifies to $$2.4x + 4 = -8$$. Now the variable appears only on the left.

**Step 2 — Remove the constant from the variable side:** The constant $$4$$ is on the variable side, so subtract $$4$$ from both sides:

$$2.4x + 4 - 4 = -8 - 4$$

$$2.4x = -12$$

**Step 3 — Isolate the variable using the Division Property of Equality:** Because $$x$$ is multiplied by the decimal coefficient $$2.4$$, divide both sides by $$2.4$$:

$$\frac{2.4x}{2.4} = \frac{-12}{2.4}$$

$$x = -5$$

**Step 4 — Check by substitution:** Replace $$x$$ with $$-5$$ in the original equation:

$$7.8(-5) + 4 \stackrel{?}{=} 5.4(-5) - 8$$

$$-39 + 4 \stackrel{?}{=} -27 - 8$$

$$-35 = -35 \checkmark$$

Because both sides are equal, $$x = -5$$ is confirmed as the correct solution. This example demonstrates that the collecting technique applies identically when the coefficients are **decimals** rather than integers or fractions. Subtracting decimal coefficients ($$7.8 - 5.4 = 2.4$$) and dividing by a decimal ($$-12 \div 2.4 = -5$$) follow the same arithmetic rules used for whole numbers, so the overall strategy remains unchanged.

Solving $$7.8x + 4 = 5.4x - 8$$ by Collecting Variables and Constants

A payroll specialist is adjusting two employee records that currently have the same total hours worked. If the specialist subtracts the same amount of unpaid break time from both records, which mathematical property justifies that the remaining hours for both employees are still equal?

In a retail inventory system, if two store locations have the same number of units in stock and both sell the exact same number of units today, their remaining stock levels will still be equal. This is a real-world application of the ____ Property of Equality.

In a payroll system, if two accounts have equal balances and the same deduction is made from both, which mathematical property justifies that the remaining balances are still equal?

Mathematical Properties in Payroll

In a corporate payroll system, if two employees have the same gross pay and the same amount is deducted from both for health insurance, the Subtraction Property of Equality justifies that their remaining net pay amounts will still be equal.

In a corporate accounting or data management environment, maintaining the balance of equations is a fundamental task. Match each aspect of the Subtraction Property of Equality with its corresponding description or representation.

A database administrator is applying the Subtraction Property of Equality to ensure two synchronized data records remain balanced after a deletion. Arrange the logical steps of this property in the correct sequence to show how the equality is preserved.

Inventory Synchronization in Logistics

In a corporate setting, maintaining consistency across financial records is essential for accurate auditing. Imagine two department managers who have been allocated exactly the same quarterly budget. At the end of the quarter, both departments are required to contribute the same fixed amount toward a company-wide technology upgrade fee. Even after this deduction, their remaining budget balances will still be equal. Identify the specific mathematical property that justifies why these balances remain equal. In your response, provide the formal algebraic definition of this property (using variables such as a, b, and c) and explain in your own words why this property is fundamental for keeping an equation balanced when solving for a variable.

Applying the Subtraction Property of Equality in Corporate Budgeting

A compliance officer is auditing two department accounts with equal balances, represented by the equation $d_1 = d_2$. If both accounts are charged an identical processing fee ($s$), which equation correctly applies the Subtraction Property of Equality to show that the remaining balances are still equal?

A facilities manager is overseeing two office buildings that currently have an equal number of workstations. If the manager removes the exact same number of workstations from both buildings during a renovation, which statement is true according to the Subtraction Property of Equality?

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