Question

In the equation \(b=k a+3, k\) is a constant. If the possible solutions are in the form \((a, b),\) is (2,3) a solution to the equation?

Short answer

Yes, (2, 3) is a solution.

Step-by-step solution

- Substitute the given values

Substitute the values of the given point \(a = 2, b = 3\) into the equation \(b = ka + 3\). This gives \(3 = k(2) + 3\).

- Simplify the equation

Rewrite the equation from Step 1: \(3 = 2k + 3\).

- Solve for k

Subtract 3 from both sides of the equation: \(3 - 3 = 2k + 3 - 3\). This simplifies to \(0 = 2k\). Divide both sides by 2: \(0 / 2 = k\). This gives \(k = 0\).

- Verify the solution

Substitute \(k = 0\) back into the original equation: \(b = 0a + 3 = 3\). Since \(b = 3\) and this matches the value of \(b\) in the given point (2, 3), (2, 3) is indeed a solution.

Key concepts

Understanding Equations

Equations are a fundamental concept in algebra. An equation shows the relationship between different variables. In our problem, we have the equation:

\[b = ka + 3\]
This equation expresses how 'b' is related to 'a' through the constant 'k'. When you understand equations well, you can solve many algebra problems more easily.

Substitution Method Explained

The substitution method is a useful technique for solving equations. Here's how it works:

• Identify the values you know.
• Replace variables in the equation with these known values.

In our problem, we know

\(a = 2\) and \(b = 3\). So, we substitute these values into the equation
\[b = ka + 3\]
and get:
\[3 = k \times 2 + 3\].
This makes the equation simpler and easier to solve.

The Role of a Constant

A constant is a fixed value that does not change. In our equation, \(k\) is the constant. Sometimes, finding the value of a constant is crucial to solving the problem.
For example, we needed to discover the value of \(k\) that makes the equation true for the given values of \(a\)and \(b\).
Using the substitution method, we converted our equation to:
\[3 = 2k + 3\]. Now, it's easier to see how \(k\)affects the equation.

Solving for k

To solve for \(k\), we need to isolate \( k \) on one side of the equation. Let's use our simplified equation:

\[3 = 2k + 3\].
First, subtract 3 from both sides:
\[3 - 3 = 2k + 3 - 3\], which simplifies to
\[0 = 2k\].
Next, divide both sides by 2: \[0/2 = k\], which gives \(k = 0\).

Finally, let's verify our solution by substituting \(k = 0 \) back into the original equation:

\[b = 0a + 3\]. Given \( a = 2 \), we get \[b = 3\]. Since \( b = 3\) matches the value in our original point, we've confirmed our solution.