In Exercises 61–66, find all values of x satisfying the given conditions. y1 = (x - 3)/5, y2 = (x - 5)/4, and y1 - y2 = 1.

Verified step by step guidance

1. Start by setting up the equation based on the given condition: $y_1-y_2=1$.
2. Substitute the expressions for $y_1$ and $y_2$ into the equation: $\frac{x-3}{5}-\frac{x-5}{4}=1$.
3. To eliminate the fractions, find a common denominator for the fractions on the left side. The common denominator of 5 and 4 is 20.
4. Rewrite each fraction with the common denominator: $\frac{4(x-3)}{20}-\frac{5(x-5)}{20}=1$.
5. Combine the fractions on the left side: $\frac{4(x-3)-5(x-5)}{20}=1$.

Key Concepts

Linear Equations

Linear equations are mathematical statements that express the equality of two linear expressions. In this context, the equations y1 and y2 represent linear functions of x. Understanding how to manipulate and solve these equations is essential for finding the values of x that satisfy the given conditions.

Substitution Method

The substitution method involves replacing one variable with an equivalent expression to simplify the problem. In this case, we can substitute the expressions for y1 and y2 into the equation y1 - y2 = 1 to find the value of x. This technique is crucial for solving systems of equations efficiently.

Solving for x

Solving for x involves isolating the variable in an equation to determine its value. This process often requires combining like terms, applying inverse operations, and simplifying expressions. In this exercise, finding the correct value of x that satisfies the equation y1 - y2 = 1 is the ultimate goal.