Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 8 x-5=3 x+10 $$

Short answer

The solution to the equation \(8x - 5 = 3x + 10\) is \(x = 3\)

Step-by-step solution

Simplify both sides of the equation by collecting the 'x' terms and constant terms

This can be done by subtracting \(3x\) from both sides of the equation and also adding \(5\) to both sides of the equation. Doing this, the equation becomes \(8x - 3x = 10 + 5 \) or \(5x = 15 \)

Solve for 'x'

Now that \(5x = 15\), to get the value of 'x', divide both sides of the equation by \(5\), thus giving \(x = 15/5\). So \(x = 3\)

Check your solution

This can be done by substituting the value of 'x' back into the original equation. Therefore, substituting \(3\) into the original equation makes it \(8*3 - 5 = 3*3 + 10\). Simplifying both sides yields \(24 - 5 = 9 + 10\) or \(19 = 19\). Since both sides are equal, the solution is correct.

Key concepts

Algebraic Equations

In mathematics, an algebraic equation is a statement of equality involving variables and constants. The objective is to find the values for the variables that make the equation true. In the case of the equation 8x - 5 = 3x + 10, x is the variable, and this equation is a classic example of a linear equation, where the variable is raised to the power of one.

Linear equations are fundamental in algebra and form the basis for more complex equations. They can always be rearranged into a standard form, ax + b = 0, where a and b are constants. To solve such an equation, we combine like terms and isolate the variable on one side of the equation, providing a clear path to the solution.

Equation Solving Strategies

Effective equation solving strategies are critical in algebra. Firstly, simplify the equation by combining like terms and using properties of equality: addition, subtraction, multiplication, and division. This step aims to transform the equation into a simpler equivalent, where the variable is easy to isolate.

Once simplified, use inverse operations to solve for the variable. In the equation provided, after combining like terms, we get 5x = 15. Here, the inverse operation is division, as the variable is multiplied by 5. Dividing both sides by 5 gives the solution for x. Always ensure each operation is performed on both sides to maintain the equation's balance.

Checking Solutions in Algebra

The final and crucial step in solving algebraic equations is checking the solutions. This ensures the solution satisfies the original equation. To check, substitute the solution back into the original equation.

Using the solution x = 3 from our exercise, replace x with 3: 8(3) - 5 on the left and 3(3) + 10 on the right. Simplify both expressions to verify they are equal, which confirms that the solution is correct. Consistently checking solutions not only validates your answer but also reinforces understanding and helps to identify calculation errors.