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Chapter 15: Problem 60

Solve by graphing. $$ x-7=2 x+5 $$

Short Answer

Expert verified
The solution is x = -12.

Step by step solution

01

Rearrange the Equation

Move all terms involving x to one side of the equation and constants to the other side. Given the equation is \(x - 7 = 2x + 5\), subtract x from both sides: \[ -7 = x + 5 \]. Next, subtract 5 from both sides: \[ -12 = x \]. So, we have \[ x = -12 \].
02

Graph the Equation

Plot the line corresponding to the left-hand side (LHS) of the original equation, y = x - 7, on a graph. Similarly, plot the line corresponding to the right-hand side (RHS) of the original equation, y = 2x + 5.
03

Identify the Intersection Point

The solution to the equation \(x - 7 = 2x + 5\) corresponds to the x-value of the intersection point of the two lines. The lines will intersect at x = -12. The coordinates of the intersection point are (-12, y) where y is the value satisfying both equations.

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Key Concepts

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

linear equation
A linear equation is an equation that describes a straight line when graphed. The equation typically takes the form: y = mx + b, where
  • m is the slope of the line
  • b is the y-intercept, or where the line crosses the y-axis.
In our example, we have two equations:
  • y = x - 7
  • y = 2x + 5
These are both linear equations because if we draw them on a graph, they'll form straight lines.
rearranging equations
Rearranging equations involves manipulating the equation to isolate one variable. In this case, we want to solve for x. Starting with the given equation:
  • x - 7 = 2x + 5
We first move all terms involving x to one side by subtracting x from both sides:
  • -7 = x + 5
Next, we isolate x by subtracting 5 from both sides:
  • -12 = x
This tells us that x = -12. Rearranging is crucial because it simplifies the equation, making it easier to find the solution.
graphing equations
Graphing equations is a visual way to solve them. To do this, we plot each equation on a graph. For our example:
  • y = x - 7
  • y = 2x + 5
We plot these equations and look for their intersection point. The x-coordinate of this point is the solution to our original equation. In our case:
  • The lines intersect at x = -12
  • The intersection is the point (-12, y)
By graphing, we see where the solutions of both equations overlap. This visual method helps us easily identify the solution.

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Most popular questions from this chapter

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Solve each system graphically. $$ \begin{array}{l} {y=\frac{1}{3} x+2} \\ {y=\frac{1}{3} x-7} \end{array} $$

Remove parentheses and simplify. $$ 4 r-(3 r+5) $$

Find two numbers for which the sum is 108 and the difference is \(50 .\)
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