To find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the given identity [tex]\(a(4x - 10) \equiv 24x + 5b\)[/tex], we need to ensure that both expressions are equivalent for all values of [tex]\(x\)[/tex]. This means that the coefficients of [tex]\(x\)[/tex] and the constant terms on both sides of the identity must be the same. Let's proceed step-by-step:
1. Expand the left-hand side (LHS):
[tex]\[
a(4x - 10) = 4ax - 10a
\][/tex]
2. Compare the coefficients of [tex]\(x\)[/tex]:
The expression on the right-hand side (RHS) is given as:
[tex]\[
24x + 5b
\][/tex]
On the LHS, the coefficient of [tex]\(x\)[/tex] is [tex]\(4a\)[/tex]. On the RHS, the coefficient of [tex]\(x\)[/tex] is [tex]\(24\)[/tex]. To make the coefficients equal, we set:
[tex]\[
4a = 24
\][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{24}{4} = 6
\][/tex]
3. Compare the constant terms:
On the LHS, the constant term is [tex]\(-10a\)[/tex]. On the RHS, the constant term is [tex]\(5b\)[/tex]. To make these terms equal, we set:
[tex]\[
-10a = 5b
\][/tex]
Substitute [tex]\(a = 6\)[/tex] into the equation:
[tex]\[
-10(6) = 5b
\][/tex]
Simplifying this:
[tex]\[
-60 = 5b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = \frac{-60}{5} = -12
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
\begin{array}{l}
a = 6 \\
b = -12
\end{array}
\][/tex]
