For which real value of x do the linear expressions 7x + 13 and 13x − 7 become equal to each other?

Introduction / Context:
This is a basic algebra question about solving a simple linear equation. The equality of two linear expressions is used to find the value of x. Such questions test your ability to rearrange and simplify equations to isolate the variable and are a foundation for more advanced algebra topics.

Given Data / Assumptions:

• The two expressions are 7x + 13 and 13x − 7.
• They are equal at some real value of x.
• We must solve 7x + 13 = 13x − 7 for x.
• Standard properties of equality and basic arithmetic operations apply.

Concept / Approach:
To solve an equation like 7x + 13 = 13x − 7, we bring all terms involving x to one side and constant terms to the other side. This gives a simple form kx = constant, from which x is easily obtained by dividing. Throughout the process, whatever operation is done to one side of the equation must be done to the other side to preserve equality.

Step-by-Step Solution:
Start with the equation 7x + 13 = 13x − 7.Subtract 7x from both sides to collect x terms on the right: 13 = 6x − 7.Add 7 to both sides to move constants to the left: 13 + 7 = 6x.Compute 13 + 7 = 20, so 20 = 6x.Divide both sides by 6: x = 20 / 6 = 10 / 3 after simplifying the fraction by dividing numerator and denominator by 2.

Verification / Alternative check:
Substitute x = 10/3 into both expressions. For 7x + 13: 7 * (10/3) + 13 = 70/3 + 13 = 70/3 + 39/3 = 109/3. For 13x − 7: 13 * (10/3) − 7 = 130/3 − 7 = 130/3 − 21/3 = 109/3. Since both expressions give the same value 109/3 when x = 10/3, the solution is correct.

Why Other Options Are Wrong:

• For x = −10/3, 7x + 13 and 13x − 7 produce different values.
• For x = 3/10 or −3/10, the values of the expressions are not equal.
• For x = 0, 7x + 13 equals 13 and 13x − 7 equals −7, which clearly are not equal.
• Only x = 10/3 satisfies the equation.

Common Pitfalls:

• Moving terms across the equals sign without changing their sign correctly.
• Making arithmetic errors while adding or subtracting constants like 13 and 7.
• Failing to simplify the final fraction 20/6 to 10/3, though even 20/6 would still represent the same numeric value.

Final Answer:
10/3