Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$5 x+8=9 x$$

Short answer

x = 2

Step-by-step solution

Isolate the variable on one side

To solve the equation for x, we need to get all x terms on one side and constant terms on the other side. We can do this by subtracting 5x from both sides of the equation.

Simplify both sides of the equation

After subtracting 5x from both sides, we get 8 = 4x. Now divide both sides by 4 to solve for x.

Divide to find the value of x

Dividing both sides of the equation by 4 gives us x = 2.

Check the solution

To check the solution, substitute x = 2 back into the original equation: 5(2) + 8 should equal 9(2). Simplify both sides to verify they are equal.

Key concepts

Isolate the Variable

Isolating the variable is the cornerstone of solving linear equations. The goal is to get the variable on one side of the equation so that its value becomes clear. When faced with an equation like \(5x + 8 = 9x\), the immediate task is to gather all terms containing the variable on one side. In this scenario, subtracting \(5x\) from both sides ensures that x is no longer on the left side. This leaves us with \(8 = 4x\).

One useful tip for visual learners is to draw a balance scale in their notes, placing the variable terms on one side and constants on the other. This helps to conceptualize the equation as a balance that must be maintained equally on both sides when manipulating it.

Simplifying Equations

The process of simplifying an equation involves breaking it down into the simplest terms possible. This makes it easier to identify the solution. Once we've isolated the variable, like in the equation \(8 = 4x\), we can simplify further by dividing both sides by the coefficient of the variable, which in this case, is 4. Simplification leads to \(x = 2\), providing us with the value of the variable.

Remember that simplification is not just about making a side of the equation equal to the variable; it's about making the equation clearer and more straightforward to solve. To aid clarity, always perform the same operation on both sides of the equation, and simplify any fractions or complex expressions where possible.

Checking Solutions

After finding a potential solution, it's crucial to validate it by substituting the value back into the original equation. For the equation \(5x + 8 = 9x\), when we substitute \(x = 2\), it transforms into \(5(2) + 8 = 9(2)\). Simplifying both sides gives us 18, confirming that our solution \(x = 2\) is correct since both sides of the original equation are now equal.

This process is likened to retracing your steps to ensure no mistakes were made along the way. A correct solution will always satisfy the original equation, a principle that holds true for all linear equations. Students can boost their confidence in solving equations by always remembering to perform this crucial step.