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Chapter 15: Problem 64

Solve by graphing. $$ 2 x-5=8 x $$

Short Answer

Expert verified
The solution to the equation is \(x = -\frac{5}{6}\).

Step by step solution

01

Rewrite the Equation

Start by rewriting the given equation in the form of a linear equation. Given: \[ 2x - 5 = 8x \] We can rewrite this as: \[ 2x - 8x - 5 = 0 \] Combine like terms to get: \[ -6x - 5 = 0 \]
02

Solve for x

Isolate the variable x by adding 5 to both sides of the equation: \[ -6x = 5 \] Next, divide both sides by -6: \[ x = -\frac{5}{6} \]
03

Graph the Equation

The equation can be expressed as two linear equations: \[ y = 2x - 5 \] and \[ y = 8x \] To find the point of intersection, simply graph these two lines on the coordinate plane.
04

Find the Intersection

The point of intersection of the lines \(y = 2x - 5\) and \(y = 8x\) represents the solution to the original equation. You will find that the lines intersect at \(x = -\frac{5}{6}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear equations
Linear equations are a fundamental part of algebra and are crucial for understanding more complex mathematical concepts. A linear equation is an equation of the form \( ax + b = c \). These equations will always graph as straight lines on the coordinate plane. Some key characteristics of linear equations are:
  • The highest power of the variable (x) is 1.
  • They can typically be written in the form \( y = mx + b \), where m is the slope and b is the y-intercept.

In our original exercise, we started with \( 2x - 5 = 8x \). Through rearranging the terms and simplifying, we got it in a more familiar linear form where comparing it to \( y = mx + b \) is easier.
graphing
Graphing is a visual way to solve equations and understand their behavior. When solving linear equations by graphing, you plot the given equations on the coordinate plane and look for points where the lines intersect. Here's a step-by-step approach to graphing:
  • Find at least two points for each equation using different x-values and calculate the corresponding y-values.
  • Plot these points on the graph.
  • Draw a straight line through the points for each equation.

In our solution, we expressed the equation as two linear equations \( y = 2x - 5 \) and \( y = 8x \). We then graphed these lines to find where they intersect. The intersection point represents the solution to the original equation.
solving equations
Solving equations is about finding the value(s) of the variable that make the equation true. This can be done algebraically or graphically. Here's a quick overview of the algebraic method we used in the given problem:
  • First, we isolated all x-terms on one side of the equation: \( 2x - 8x - 5 = 0 \).
  • Next, we combined like terms: \( -6x - 5 = 0 \).
  • Finally, we solved for x by isolating it: \( -6x = 5 \) and then \( x = -\frac{5}{6} \).

By graphing the linear equations \( y = 2x - 5 \) and \( y = 8x \), we confirmed that both algebraic and graphical methods yield the same solution, \( x = -\frac{5}{6} \). This dual approach reinforces your understanding and ensures accuracy.

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Most popular questions from this chapter

The Nuthouse has 10 kg of mixed cashews and pecans worth \(\$ 8.40\) per kilogram. Cashews alone sell for \(\$ 8.00\) per kilogram, and pecans sell for \(\$ 9.00\) per kilogram. How many kilograms of each nut are in the mixture?

Solve each system using the elimination method. $$ \begin{aligned} &3 x+y=-1\\\ &2 x-3 y=-8 \end{aligned} $$

Translate each problem to an equation. Do not solve. The average annual cost for badminton is \(\$ 12 .\) This is \(\$ 458\) less than the average annual cost for scuba diving. What is the average annual cost for scuba diving?

Find the zero of each function. $$ f(x)=2 x+9 $$

Combine like terms. $$ -6 x+7+9 x $$
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