To determine the value of [tex]\(a\)[/tex] that makes the equation have a solution for all real numbers, we need to ensure that both sides of the equation can be made equivalent for any value of [tex]\(x\)[/tex].
Given the equation:
[tex]\[2x - ax - 15 = a(2x - 3) - 13x\][/tex]
### Step 1: Expand both sides
1. Left Side:
[tex]\[2x - ax - 15\][/tex]
2. Right Side:
[tex]\[a(2x - 3) - 13x\][/tex]
Distribute [tex]\(a\)[/tex] on the right:
[tex]\[2ax - 3a - 13x\][/tex]
So, the equation now looks like:
[tex]\[2x - ax - 15 = 2ax - 3a - 13x\][/tex]
### Step 2: Group similar terms for comparison
For the equation to be true for all values of [tex]\(x\)[/tex], the coefficients of [tex]\(x\)[/tex] and the constant terms on both sides must be equal. Therefore, we need to compare:
1. Coefficients of [tex]\(x\)[/tex]:
Comparing the coefficients of [tex]\(x\)[/tex] on both sides:
[tex]\[2 - a = 2a - 13\][/tex]
2. Constant Terms:
Comparing the constant terms:
[tex]\[-15 = -3a\][/tex]
### Step 3: Solve for [tex]\(a\)[/tex]
1. Solve the constant terms equation first:
[tex]\[-15 = -3a\][/tex]
Divide both sides by [tex]\(-3\)[/tex]:
[tex]\[a = \frac{-15}{-3} = 5\][/tex]
2. Validate by solving the coefficients equation:
Substitute [tex]\(a = 5\)[/tex] into the coefficients equation:
[tex]\[2 - a = 2a - 13\][/tex]
[tex]\[2 - 5 = 2(5) - 13\][/tex]
Simplify both sides:
[tex]\[-3 = 10 - 13\][/tex]
[tex]\[-3 = -3\][/tex]
Since both sides are equal, the value [tex]\(a = 5\)[/tex] is correct.
Hence, the value of [tex]\(a\)[/tex] that makes the equation true for all real numbers is:
[tex]\[ \boxed{5} \][/tex]
