Question

If \(y=\frac{5}{3} x^{3}+2 x+C\) and \(y=5\) when \(x=3,\) find the value of \(C\).

Short answer

The value of \(C\) is \(-46\).

Step-by-step solution

- Substitute given values into the equation

Substitute the given values of \(x = 3\) and \(y = 5\) into the equation \(y = \frac{5}{3} x^3 + 2x + C\). This gives us the equation \(5 = \frac{5}{3}(3)^3 + 2(3) + C\).

- Simplify the equation

Calculate the value of \( \frac{5}{3}(3)^3 \) and \(2(3)\). This simplifies to \(5 = \frac{5}{3} \cdot 27 + 6 + C\), which further simplifies to \(5 = 45 + 6 + C\).

- Solve for C

Combine like terms to get \(5 = 51 + C\). Then, isolate \(C\) by subtracting 51 from both sides, yielding \(C = 5 - 51\). Therefore, \(C = -46\).

Key concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In algebra, we use letters, known as variables, to represent unknown values or values that can change. This provides a powerful way to solve real-world problems by setting up equations and solving for these variables. For example, in the problem given, we use the variable \(x\) to represent input values and \(C\) to represent an unknown constant. The goal is to find the value of \(C\) using algebraic techniques.

Equations

An equation is a mathematical statement that asserts the equality of two expressions. It consists of two parts separated by an equals sign (\( = \)). In our exercise, we started with the equation \( y = \frac{5}{3} x^3 + 2x + C \). When we are given specific values for \(x\) and \(y\), we can substitute these into the equation to solve for the unknown variable, in this case, the constant \(C\).
Breaking down this process:

• First, substitute the values. For \(x = 3\) and \(y = 5\), the equation becomes \( 5 = \frac{5}{3}(3)^3 + 2(3) + C \).
• Next, perform the calculations. Simplify \( \frac{5}{3} \times 27 \) and \( 2 \times 3 \) to get \( 5 = 45 + 6 + C \).
• Combine like terms: \(5 = 51 + C\).
• Isolate \(C\) by subtracting 51 from both sides: \(C = 5 - 51\).
• Thus, \( C = -46 \).

Constants

Constants in mathematics are values that do not change. Unlike variables, which can represent various numbers, constants have a fixed value. In the problem provided, \(C\) is a constant that we needed to determine.
To do this, we substituted the values for \(x\) and \(y\) into the given equation and solved for \(C\). After simplifying the equation, we found \(C = -46\). This value remains the same for the given equation under the provided conditions. Understanding constants is crucial, as they often represent fixed quantities in various mathematical contexts.