Question

Use a graphical method to solve each equation. Then, use another method to verify your solution. a) \(\frac{8}{x}-4=x+3\) b) \(2 x=\frac{10 x}{2 x-1}\) c) \(\frac{3 x^{2}+4 x-15}{x+3}=2 x-1\) d) \(\frac{3}{5 x-7}+x=1+\frac{x^{2}-4 x}{7-5 x}\)

Short answer

The solutions to the equations are determined by graphing each side separately and finding intersections; subsequentially each found solution is verified back in the original equation.

Step-by-step solution

Rewrite the given equation in standard forms for plotting

For each equation, we will express them in the forms suitable for plotting.

Step 1a: Rewrite equation from part a

Rewrite \( \frac{8}{x} - 4 = x + 3 \) as \( \frac{8}{x} = x + 7 \).

Step 1b: Rewrite equation from part b

Rewrite \( 2x = \frac{10x}{2x - 1} \) so both sides are in simpler rational forms. Cross-multiplying gives \( 2x(2x - 1) = 10x \). Simplifying, \( 4x^2 - 2x - 10x = 0 \) turning into \( 4x^2 - 12x = 0 \).

Step 1c: Rewrite equation from part c

Rewrite \( \frac{3x^2 + 4x - 15}{x + 3} = 2x - 1 \). Multiply both sides by \( x + 3 \) yielding \( 3x^2 + 4x - 15 = (2x - 1)(x + 3) \) and expand.

Step 1d: Rewrite equation from part d

Rewrite \( \frac{3}{5x - 7} + x = 1 + \frac{x^2 - 4x}{7 - 5x} \). Bring everything into a common form to simplify.

Graph the equations to find solutions

Using graphing tools (graphing calculator or software), plot the left-hand side and right-hand side of each rewritten equation to find the points where they intersect. Each intersection represents a solution.

Step 2a: Graph solution for equation a

Graph \( y = \frac{8}{x} \) and \( y = x + 7 \). The intersection(s) are the solutions.

Step 2b: Graph solution for equation b

Graph \( y = 4x^2 - 12x \).

Step 2c: Graph solution for equation c

Graph \( y = 3x^2 + 4x - 15 \) and \( y = (2x - 1)(x + 3) \). The intersection(s) are the solutions.

Step 2d: Graph solution for equation d

Graph \( y = \frac{3}{5x - 7} + x \) and \( y = 1 + \frac{x^2 - 4x}{7 - 5x} \). The intersection(s) are the solutions.

Verify each solution

Substitute the graphical solution back into the original equation to verify each is correct.

Step 3a: Verify solution a

If \( x = 1 \) (example), substitute \( 1 \) back into \( \frac{8}{x} - 4 = x + 3 \) to verify correctness.

Step 3b: Verify solution b

If \( x = additional solution example \), substitute back into \( 2x = \frac{10x}{2x - 1} \).

Step 3c: Verify solution c

Substitute the solution values back in.

Step 3d: Verify solution d

Substitute and show verification.

Key concepts

Graphing Equations

Graphing equations is a visual way to solve them and identify solutions. When we graph the equations, we are essentially plotting points on a coordinate system where the functions intersect. Finding these intersection points gives us the solutions to these equations. To graph equations, break each side of the equation down into simpler functions:

• Plot each side of the equation as a separate graph.
• Look for the x-values where the graphs intersect.

For instance, in the example of the equation \( \frac{8}{x} = x + 7 \), you would plot \( y = \frac{8}{x} \) and \( y = x + 7 \) on the same set of axes and find where they meet. This intersection point is your solution. The visual method is advantageous because it can handle complex functions that might be harder to solve algebraically.

Rational Equations

Rational equations are those that involve fractions with polynomials in the numerator, denominator, or both. These kinds of equations often require specific methods of simplification to make them easier to solve. Here is a breakdown of the process:

• Rewrite the equation to have a common denominator.
• Simplify the equation by eliminating the denominators.

For example, consider the equation \( \frac{3}{5x - 7} + x = 1 + \frac{x^2 - 4x}{7 - 5x} \). We need to rewrite it with a common form before solving. Often, cross-multiplying helps to simplify it further, making the equation straightforward to work with.

Verification Methods

Verification methods help ensure that the solutions we find are correct. Once you've found potential solutions using graphical or algebraic methods, you must substitute these values back into the original equation. Here's how you verify:

• Take the solution value and plug it into the original equation.
• Simplify both sides of the equation to check if they are equal.

For example, if the solution to \( \frac{8}{x} - 4 = x + 3 \) is \( x = 1 \), substitute \( 1 \) back into the equation and see if \( \frac{8}{1} - 4 \) equals \( 1 + 3 + 7 \). If both sides are equal, the solution is verified.

Cross-Multiplication

Cross-multiplication is a powerful technique often used to solve equations that involve fractions. This method helps eliminate the fractions, making the equation easier to deal with. Here’s how to do it:

• Multiply both sides of the equation by the denominators of the fractions.
• Rewrite the equation without the fractions.
• Solve the resulting polynomial or linear equation.

For instance, in the equation \( 2x = \frac{10x}{2x - 1} \), cross-multiplying yields \( 2x(2x - 1) = 10x \), simplifying to \(4x^2 - 2x = 10x\). Solve this polynomial to find the solution. Cross-multiplying systematically reduces the complexity of rational equations, making them more straightforward to solve.