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Chapter 4: Problem 32

Solve each system of equations by graphing. If the system is inconsistent or the equations are dependent, say so. $$ \begin{aligned} &3 x=y+5\\\ &6 x-5=2 y \end{aligned} $$

Short Answer

Expert verified
The system is inconsistent and has no solution.

Step by step solution

01

Rewrite each equation in slope-intercept form

Convert each equation into the form \(y = mx + b\), which makes it easier to graph. Start with the first equation:\[3x = y + 5\]Solve for \(y\):\[y = 3x - 5\]
02

Rewrite the second equation in slope-intercept form

Now consider the second equation:\[6x - 5 = 2y\]Solve for \(y\):\[2y = 6x - 5\]Divide by 2:\[y = 3x - \frac{5}{2}\]
03

Graph the first equation

Graph the equation \[y = 3x - 5\]. Use the slope \(m = 3\) and the y-intercept \(b = -5\). Plot the y-intercept on the y-axis, then use the slope to find another point and draw the line.
04

Graph the second equation

Graph the equation \[y = 3x - \frac{5}{2}\]. Use the slope \(m = 3\) and the y-intercept \(b = -\frac{5}{2}\). Plot the y-intercept on the y-axis, then use the slope to find another point and draw the line.
05

Analyze the graphs

Examine the graphs of both lines. Since both lines have the same slope but different y-intercepts, they are parallel. Therefore, the system of equations is inconsistent and there is no solution.

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Key Concepts

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

Graphing Linear Equations
Graphing is a visual way to represent equations on a coordinate plane. When solving a system of equations by graphing, you plot each equation as a line and then look for points where the lines intersect. Here are steps you generally follow:
  • Reformat the equation into a graph-ready form, usually slope-intercept form.
  • Identify key features like the slope and y-intercept.
  • Plot the y-intercept on the y-axis.
  • Use the slope to determine additional points on the line.
  • Draw the line through these points.
Let鈥檚 remember that intersections between lines indicate solutions common to both equations. No intersection suggests no solutions, and overlapping lines indicate infinite solutions.
Slope-Intercept Form
The slope-intercept form of a line is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) represents the y-intercept. This form is particularly useful because it gives direct insight into the line's orientation and starting point.
To convert an equation to slope-intercept form, isolate \(y\) on one side of the equation. Let鈥檚 see how it applies to our example:
For the first equation \[3x = y + 5\], solve for \(y\):
\

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Most popular questions from this chapter

If \(x\) units of a product cost \(C\) dollars to manufacture and earn revenue of \(R\) dollars, the value of \(x\) at which the expressions for \(C\) and \(R\) are equal is called the break-even quantity-the number of units that produce 0 profit. (a) find the break-even quantity, and (b) decide whether the product should be produced on the basis of whether it will earn a profit. (Profit\(=\)Revenue\(-\)cost) \(C=85 x+900 ; R=105 x\) No more than 38 units can be sold.

Write a system of equations for each problem, and then solve the system. RAGBRAI", the Des Moines Register's Annual Great Bicycle Ride Across Iowa, is the longest and oldest touring bicycle ride in the world. Suppose a cyclist began the \(471 \mathrm{mi}\) ride on July \(20,2008,\) in western Iowa at the same time that a car traveling toward it left eastern Iowa. If the bicycle and the car met after 7.5 hr and the car traveled \(35.8 \mathrm{mph}\) faster than the bicycle, find the average rate of each.

Solve each system by the substitution method. Check each solution. $$ \begin{aligned} &\frac{1}{2} x-\frac{1}{8} y=-\frac{1}{4}\\\ &\frac{1}{3} x-\frac{1}{12} y=-\frac{1}{6} \end{aligned} $$

Without graphing, answer the following questions for each linear system. (a) Is the system inconsistent, are the equations dependent, or neither? (b) Is the graph a pair of intersecting lines, a pair of parallel lines, or one line? (c) Does the system have one solution, no solution, or an infinite number of solutions? $$ \begin{aligned} &y+2 x=6\\\ &x-3 y=-4 \end{aligned} $$

Solve each system by the elimination method. $$ \begin{array}{c} {5 x-2 y=3} \\ {10 x-4 y=5} \end{array} $$
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