Question

Use a graphing calculator to find the solution of the equation. Check your solution algebraically. $$\frac{1}{3}(10 x+27)=-\frac{1}{5}\left(\frac{55}{3} x+25\right)$$

Short answer

The solution to this equation is \(x = -2\). This can be checked by substituting the value of \(x\) back into the original equation. It would be helpful to use a graphing calculator to graph the equation and confirm the solution graphically as well.

Step-by-step solution

Simplify the equation

Simplify the equation as follows: \[\frac{1}{3}(10x+27) = -\frac{1}{5}\left(\frac{55}{3}x+25\right)\] Multiply each side of the equation by 15, the common multiple of 3 and 5, to eliminate fractions. This simplifies the equation to: \[5(10x+27) = -3\left(\frac{55}{3}x+25\right)\] This further simplifies to: \[50x+135 = -55x -75\]

Collect Like Terms

Collect all terms involving x on one side and constants on the other by adding \(55x\) and \(75\) to both sides of the equation: \[50x + 55x = -75 -135\] Which simplifies to: \[105x = -210\]

Solve for X

You can now solve for x by dividing both sides of the equation by 105: \[x = \frac{-210}{105}\] Which simplifies to: \[x = -2\]

Key concepts

Graphing Calculator

A graphing calculator is a high-powered calculator with advanced functionalities, including the ability to graph equations. It is a handy tool for visualizing the relationship between variables in an equation. To use a graphing calculator to find the solution of a linear equation, you first need to understand how to rewrite the equation in a form the calculator can interpret.

For our example equation, \(\frac{1}{3}(10x+27)=-\frac{1}{5}\)\(\left(\frac{55}{3}x+25\right)\), you would input the left and right sides as separate functions and then use the graphing feature to see where these two functions intersect on the x-axis. The x-coordinate at the point of intersection represents the solution to the equation. After finding the approximate solution using the graphing calculator, you should always verify the solution algebraically to ensure accuracy.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to understand or solve. This problem-solving technique is fundamental in algebra and helps you isolate the variable of interest. In the step-by-step solution provided, algebraic manipulation included multiplying both sides of the equation by a common factor to eliminate fractions, then combining like terms, and finally, isolating the variable by dividing both sides of the equation by a coefficient.

Understanding and mastering algebraic manipulation is crucial as it forms the basis for solving more complex equations and problems. It's a tool that aids in simplifying expressions and solving for unknown values proficiently.

Collecting Like Terms

Collecting like terms is a technique used in algebra to simplify equations by combining terms that have the same variables raised to the same power. In our exercise, after eliminating fractions and expanding, we were left with terms containing the variable 'x' on both sides of the equation. By collecting like terms, which meant moving all the 'x' terms to one side and the constant terms to the other, we simplified the equation to a single variable form, \(105x = -210\).

This step is essential as it sets the stage for isolating the variable and finding the solution. When collecting like terms, always pay attention to the signs (positive or negative) to ensure accuracy in your solution.

Eliminating Fractions

Eliminating fractions from an equation can simplify the process of solving it. One common method to eliminate fractions is to multiply each term by the least common denominator (LCD). In our example, the fractions had denominators of 3 and 5. The LCD of 3 and 5 is 15, so we multiplied every term by 15 to remove the fractions and simplify the equation.

Eliminating fractions early in the solving process makes it easier to see the 'big picture' of the equation by converting it into a simple integer-based equation. Remember, whatever you do to one side of the equation, you must do to the other to maintain balance. This principle ensures that the equality remains true throughout the manipulation process.