Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$w+3=10$$

Short answer

The solution to the equation w + 3 = 10 is w = 7.

Step-by-step solution

Isolate the variable

Subtract 3 from both sides of the equation to isolate the variable w.

Simplify the equation

After subtracting 3 from both sides, the equation simplifies to w = 10 - 3.

Calculate the value of w

Perform the subtraction on the right side to find the value of w, which is w = 7.

Check the solution

To check the solution, substitute the value of w back into the original equation: (7) + 3 = 10. Verify if the left side equals the right side.

Key concepts

Algebraic Equations

An algebraic equation is like a scale where the weights on both sides are in balance. The most fundamental skill in algebra is to manipulate these equations to find the value of an unknown variable. In this case, the equation is in the form of \( w + 3 = 10 \). Think of the equation as a seesaw, with \(w + 3\) on one side and \(10\) on the other.

When solving, we aim to find the value of \(w\) that will keep the seesaw balanced. To maintain this balance, whatever you do to one side of the equation, you must do to the other. This principle keeps the two sides equal as we work toward isolating the variable to find its exact value.

Isolating Variables

Isolating the variable means rearranging the equation so that the variable we are solving for, in this case \(w\), is alone on one side of the equation. To keep the 'seesaw' balanced, subtraction was used here as a tool to eliminate the \(+3\) on the left side. By subtracting \(3\) from both sides, we achieved the equation \(w = 7\).

It is crucial to remember that isolating the variable is just one part of the process. In more complex equations, you might have to perform multiple operations, like addition, subtraction, multiplication, or division, to get the variable by itself. The goal is always to reverse the operations that are affecting the variable, peeling them away until the variable stands alone.

Fractional Solutions

In mathematics, precision is key. When dealing with exact numbers, a fractional solution is often preferable to a decimal because it is more accurate. For example, instead of saying that half of something is \(0.5\), we say it is \(\frac{1}{2}\). The latter represents the exact value.

In the given problem, the solution was simple enough that it resulted in a whole number, \(w = 7\). However, in cases where we get answers like \(\frac{8}{3}\), keeping the answer in fractional form is essential for exact calculations. Always pay attention to the instructions given; if a problem asks for a fractional answer, ensure that you leave your result in fraction form. This preserves the precision of your result, which is particularly important in higher-level mathematics or when the solution will be used in further calculations.