Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$15 x-9=7 x-5$$

Short answer

\(x = \frac{1}{2}\)

Step-by-step solution

Isolate the variable on one side

To solve the equation, start by bringing all the terms containing the variable on one side and the constants on the other side. Subtract 7x from both sides of the equation to move the terms with variables to the left side, and add 9 to both sides to move the constants to the right side. This results in: \(15x - 7x = -5 + 9\).

Simplify both sides

Combine like terms on both sides of the equation to simplify it. On the left side, combine the x terms, and on the right side, combine the constant numbers: \(8x = 4\).

Divide by the coefficient of x

To find the value of x, divide both sides of the equation by the coefficient of x, which is 8: \(x = \frac{4}{8}\).

Simplify the fraction

Simplify the resulting fraction by dividing the numerator and the denominator by their greatest common divisor, which is 4: \(x = \frac{4\div 4}{8\div 4} = \frac{1}{2}\).

Check the solution

Substitute the found value of x back into the original equation to check if it satisfies the equation: \(15\left(\frac{1}{2}\right) - 9\) should equal \(7\left(\frac{1}{2}\right) - 5\). Simplify both sides to check for equality.

Key concepts

Isolating the Variable

When solving linear equations, a critical step is to 'isolate the variable'—in other words, to get the variable by itself on one side of the equation. To achieve this, you'll need to perform operations that will move all terms containing the variable to one side and the constant terms to the other. This often involves using the properties of equality, which state that you can do the same thing (like adding, subtracting, multiplying, or dividing) to both sides of the equation without affecting its balance.

For example, if we take the equation from our exercise, subtracting 7x from both sides and adding 9 helps us isolate the variable 'x'. We end up with all the x terms on one side of the equation, setting the stage for easier manipulation and solution-finding. By successfully isolating the variable, we move one step closer to achieving the solution.

Combining Like Terms

Once you've shuffled your terms to each side of the equation, the next step in solving a linear equation is 'combining like terms'. This means you add or subtract terms that have the same variable raised to the same power. Pay attention to their coefficients as well, as they determine the value of these like terms.

In our exercise, we combined like terms by recognizing that both 15x and -7x are like terms. We combine them by subtracting the coefficients to simplify the equation to 8x on the left-hand side. This process streamlines the equation, making the subsequent steps towards finding the solution more straightforward.

Simplifying Fractions

Why Simplify Fractions?

After isolating the variable and finding its coefficient, sometimes you'll end up with a fraction. The goal then is 'simplifying fractions' to their lowest terms, which can make further calculations and your final answer much tidier. Simplification can be done by dividing the numerator and the denominator by their greatest common divisor (GCD).

In our solution process, the fraction \( \frac{4}{8} \) can be simplified because the numerator and the denominator share a GCD of 4. So, we divide both by 4 to get the fraction \( \frac{1}{2} \)—a much simpler form. This step is vital in helping us correctly interpret the result and, if necessary, check the solution in its most reduced form.

Equation Solving Step-by-Step

The concept of 'equation solving step-by-step' is a structured approach to finding an equation's solution. You advance through a series of intentional steps, each built on the last, to solve the equation methodically. The steps often start with simplifying expressions and isolating the variable, followed by operations to solve for that variable, and conclude with verifying the solution.

For instance, in the given problem, we started with rearranging terms, moved onto combining like terms, and then divided by the coefficient of the variable. Finally, we simplified the fraction and checked the solution by plugging it back into the original equation. This incremental approach not only helps to avoid small mistakes but also ensures a complete and thorough understanding of the process involved in solving equations.