Question

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

Short answer

The solution to the equation x + 9 = 5 is x = -4.

Step-by-step solution

Isolate the variable x

In order to solve the equation for the variable x, you need to isolate x on one side of the equation. To do this, subtract 9 from both sides of the equation to cancel out the +9 on the left side.

Solve for x

After subtracting 9 from both sides, the equation becomes x = 5 - 9. Then, subtract 9 from 5 to find the value of x.

Simplify the result

Simplify the subtraction on the right side of the equation to find the value of x which is x = -4.

Check the solution

To verify the solution, substitute x = -4 back into the original equation to check if both sides are equal: (-4) + 9 = 5. After performing the addition, we see that 5 = 5, which confirms the solution is correct.

Key concepts

Isolate the Variable

In algebra, the process of solving equations often begins with one fundamental step: isolating the variable. This refers to rearranging the equation so that the variable we are solving for stands alone on one side of the equality. For example, consider the equation \(x + 9 = 5\). Here, the goal is to determine the value of \(x\) that makes the equation true.

To isolate \(x\), subtract \(9\) from both sides, effectively 'moving' the \(9\) to the other side of the equation and changing its sign due to the properties of equality. The subtraction negates the addition of \(9\), leaving \(x\) by itself: \[\begin{align*} x + 9 - 9 &= 5 - 9 \ x &= -4 \end{align*}\].

By isolating the variable, we simplify the problem and take a significant step toward finding the solution. Always remember to perform the same operation on both sides of the equation to maintain balance.

Fractions in Mathematics

Fractions play a central role in mathematics, particularly in algebra. While it's often convenient to use decimals, fractions provide exact values and are sometimes preferred in mathematical solutions for their precision. When solving equations with fractions, keep in mind that the numerator represents the number of parts we have, and the denominator represents the total number of equal parts in a whole.

In some cases, when working with equations, maintaining fractional form can make the math clearer and help you avoid rounding errors that could occur with decimals. Operations with fractions follow the same fundamental principles as whole numbers. When adding or subtracting fractions, they must have a common denominator. Multiplication and division involve numerators and denominators separately. \

Example with Fractions in an Equation<\/h4>\If we have \(\frac{3}{4}x = \frac{9}{8}\), to solve for \(x\), we would multiply both sides by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\) to isolate the variable.

Checking Solutions

After solving an algebraic equation and finding the value of the variable, it's crucial to check that the solution is correct. This step ensures the original equation is indeed satisfied by the value we have found. To check the solution, we insert the calculated variable value back into the original equation and perform the operations to see if the equation holds true.

For instance, with our solution \(x = -4\), we substitute \(x\) with \( -4 \) in the original expression: \[\begin{align*} (-4) + 9 &= 5 \ 5 &= 5 \end{align*}\].

If both sides of the equation match after the substitution, we have correctly solved the problem. Ensuring both sides of the equation balance is an essential habit in mathematics as it helps prevent errors and solidifies understanding.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions, typically involving at least one variable. They can vary from simple, like \(x + 5 = 10\), to quite complex with multiple variables and various operations. The beauty of algebraic equations lies in their ability to represent real-world scenarios in an abstract manner, paving the way for a systematic approach to problem-solving.

The core principles of algebra involve combining like terms, balancing equations, and using inverse operations to simplify and solve for unknowns. Keeping equations balanced, which means performing the same operations on both sides of the equation, is essential. \

Equation Complexity and Tools<\/h4>\As equations become more complex, tools such as factoring, the quadratic formula, or rules for exponents and radicals come into play. However, the initial approach to any equation is usually to simplify gradually while adhering to the arithmetic hierarchy and aiming to isolate variables one at a time.