Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

For what value of \(c\) is the number \(a\) a solution of the equation? $$3 x-2+6 c=2 c-5 x+1 ; \quad a=4$$

Step-by-Step Solution

Answer

The value of \(c\) is \(-\frac{29}{4}\).

Step 1: Substitute the value of a

Since we are given that \(a = 4\), substitute \(x = 4\) into the equation \(3x - 2 + 6c = 2c - 5x + 1\).

Step 2: Substitute and simplify both sides

Substitute \(x = 4\) into the equation. This gives:\[3(4) - 2 + 6c = 2c - 5(4) + 1\]Simplify both sides:\[12 - 2 + 6c = 2c - 20 + 1\]This simplifies to:\[10 + 6c = 2c - 19\]

Step 3: Collect terms with c on one side

Move all terms involving \(c\) to one side of the equation and constant terms to the other:\[10 + 6c - 2c = -19\]This simplifies to:\[4c = -29\]

Step 4: Solve for c

Divide both sides by 4 to solve for \(c\):\[c = \frac{-29}{4}\]

Key Concepts

Linear Equations

Linear equations play a crucial role in algebra. They are equations that appear in the form of a straight line when plotted on a graph. These equations follow the format of:

• ax + b = 0,
• where a and b are constants and x represents a variable.

Linear equations can be simple or involve several terms both on the left and right side of the '=' sign. The purpose of solving a linear equation is to find the value of the unknown variable(s) that make the equation true.
In our original exercise, we deal with a linear equation involving the variable x and another term c. The key to solving such equations is to manipulate them algebraically to isolate the unknown variables by applying various algebraic rules like addition, subtraction, multiplication, or division.

Substitution Method

The substitution method is a common and effective technique used to solve equations. It involves replacing one variable with a given value or another expression to simplify the situation. This method is especially useful when one of the values or variables is known explicitly.
In our problem, we know that a = 4, and we use substitution by replacing x with 4 in the given equation. This permits us to break down the complexity of the equation and manage the numeric operations accurately. The goal is to reduce the unknowns step by step, leading to an eventual solution for the remaining variables like c in our case.

Solving for Variables

When solving equations, our main task is isolating the variable to find its value. After substitution in our exercise, the equation reduces to a simpler format, allowing us to rearrange terms conveniently. This makes it easier to isolate the variable we are solving for.
To solve for c, we ensure all terms involving c are on one side of the equation, with constants on the opposite side. By performing operations such as addition or subtraction, we manage to isolate c. This results in an equation involving only c, like 4c = -29.

Simplification of Equations

Simplification is essential in algebra to solve equations efficiently. It involves breaking down equations into simpler forms through basic arithmetic operations such as addition, subtraction, multiplication, and division. This reduces the complexity and makes it easier to identify solutions.
In our example, after substitution, we simplified both sides of the equation leading us to

• 10 + 6c = 2c - 19

We then re-arranged and combined like terms to further reduce the complexity to

• 4c = -29

Finally, to get the value of c, we divide both sides by 4. Simplification ensures we manage the terms systematically, reaching a clear solution quickly.