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Chapter 3: Problem 13

Solve the following systems: $$ \begin{array}{rrr} \text { a. } \frac{x}{3}+\frac{y}{2}=1 & \text { b. } \frac{x}{4}+y=9 \\ x-y=\frac{4}{3} & y=\frac{x}{2} \end{array} $$

Short Answer

Expert verified
For system a: \(x = 2\) and \(y = \frac{2}{3}\). For system b: \(x = 12\) and \(y = 6\).

Step by step solution

01

Title - Solve the first equation of system a for x

\(\frac{x}{3} + \frac{y}{2} = 1\). Multiply the entire equation by 6 to eliminate fractions: \(2x + 3y = 6\)
02

Title - Express x in terms of y for the second equation of system a

The second equation of system a is \(x - y = \frac{4}{3}\). Multiply by 3 to clear the fraction: \(3x - 3y = 4\). Rearrange to express x: \(3x = 3y + 4\), thus \(x = y + \frac{4}{3}\).
03

Title - Substitute x in the first transformed equation of system a

Substitute \(x = y + \frac{4}{3}\) into the first equation \(2x + 3y = 6\): \(2(y + \frac{4}{3}) + 3y = 6\). Simplify: \(2y + \frac{8}{3} + 3y = 6\).
04

Title - Solve for y from the substituted equation in system a

Combine like terms: \(5y = 6 - \frac{8}{3}\). Simplify the right side: \(5y = \frac{18}{3} - \frac{8}{3}\), \(5y = \frac{10}{3}\). Solve for y: \(y = \frac{2}{3}\).
05

Title - Solve for x using y in system a

Use the expression \(x = y + \frac{4}{3}\) with \(y = \frac{2}{3}\): \(x = \frac{2}{3} + \frac{4}{3} = 2\). Solution for system a: \(x = 2\) and \(y = \frac{2}{3}\).
06

Title - Substitute y in terms of x in the first equation of system b

From \(y = \frac{x}{2}\), substitute into \(\frac{x}{4} + y = 9\): \(\frac{x}{4} + \frac{x}{2} = 9\).
07

Title - Solve the substituted equation for x in system b

Combine like terms: \(\frac{x}{4} + \frac{2x}{4} = 9\), simplifying to \(\frac{3x}{4} = 9\). Multiply both sides by 4: \(3x = 36\), then solve: \(x = 12\).
08

Title - Solve for y in system b using x

Use the expression \(y = \frac{x}{2}\) with \(x = 12\): \(y = \frac{12}{2} = 6\). Solution for system b: \(x = 12\) and \(y = 6\).

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Key Concepts

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

Fraction Elimination
Fractions in equations can make solving them more challenging. To simplify our work, we often eliminate fractions. This is done by finding a common multiple, typically the least common multiple (LCM) of denominators, and multiplying every term by this multiple. Here, we can take the example from system a: \(\frac{x}{3} + \frac{y}{2} = 1\). The LCM of 3 and 2 is 6. By multiplying every term by 6, we get:
\[6 \times \frac{x}{3} + 6 \times \frac{y}{2}= 6 \times 1\].
This simplifies to: \(2x + 3y = 6\).
Notice how much easier it is to work with whole numbers instead of fractions. This technique can be applied to any linear equation that contains fractions, making it simpler and faster to find a solution.
Substitution Method
The substitution method is very useful for solving systems of linear equations. The idea is to solve one equation for one variable and then substitute that expression into the other equation. This highlights the power of substitution.
In system a, we took the second equation \(x - y = \frac{4}{3}\) and eliminated the fraction by multiplying everything by 3, resulting in \(3x - 3y = 4\).
We then solved for \(x\): \(3x = 3y + 4\), which simplifies to \(x = y + \frac{4}{3}\).
Next, we substitute this expression for \(x\) in the first equation \(2x + 3y = 6\). This substitution transforms the equation and makes it easier to solve. By substituting correctly, one of the equations is effectively removed, leaving a single-variable equation that is simpler to handle.
Linear Equations
A linear equation is any equation that can be written in the form \(ax + by = c\). Here, \(a\), \(b\), and \(c\) are constants. Linear equations represent straight lines when graphed on a coordinate plane. Understanding their properties helps to solve systems of equations efficiently.
Take system b from the exercise: \(\frac{x}{4} + y = 9\) and \(y = \frac{x}{2}\). Converting these into simpler equations without fractions: \(x + 4y = 36\) and \(2y = x\). These equations are now easier to work with. When solving a system of linear equations, you're essentially looking for a point where these lines intersect. That's the solution to the system.
In system b, substituting \(y\) from the second equation into the first gives a single equation to solve, leading to the solution for \(x\) and then \(y\).

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Most popular questions from this chapter

(Graphing program required.) Two professors from Purdue University reported that for a typical small-sized fertilizer plant in Indiana the fixed costs were \(\$ 235,487\) and it cost \(\$ 206.68\) to produce each ton of fertilizer. a. If the company planned to sell the fertilizer at \(\$ 266.67\) per ton, find the cost, \(C,\) and revenue, \(R\), equations for \(x\) tons of fertilizer. b. Graph the cost and revenue equations on the same graph and calculate and interpret the breakeven point. c. Indicate the region where the company would make a profit and create the inequality to describe the profit region.

Consider the following function: $$ f(x)=\left\\{\begin{array}{ll} 2 x+1 & x \leq 0 \\ 3 x & x>0 \end{array}\right. $$ Evaluate \(f(-10), f(-2), f(0), f(2),\) and \(f(4)\)

(Graphing program recommended.) The Ontario Association of Sport and Exercise Sciences recommends the minimum and maximum pulse rates \(P\) during aerobic activities, based on age \(A\). The maximum recommended rate, \(P_{\max },\) is \(0.87(220-A)\). The minimum recommended pulse rate, \(P_{\min },\) is \(0.72(220-A)\) a. Convert these formulas to the \(y=m x+b\) form. b. Graph the formulas for ages 20 to 80 years. Label the regions of the graph that represent too high a pulse rate, the recommended pulse rate, and too low a pulse rate. c. What is the maximum pulse rate recommended for a 20-year-old? The minimum for an 80 -year-old? d. Construct an inequality that describes too low a pulse rate for effective aerobic activity. e. Construct an inequality that describes the recommended pulse range.

In studying populations (human or otherwise), the two primary factors affecting population size are the birth rate and the death rate. There is abundant evidence that, other things being equal, as the population density increases, the birth rate tends to decrease and the death rate tends to increase. \(^{5}\) a. Generate a rough sketch showing birth rate as a function of population density. Note that the units for population density on the horizontal axis are the number of individuals for a given area. The units on the vertical axis represent a rate, such as the number of individuals per 1000 people. Now add to your graph a rough sketch of the relationship between death rate and population density. In both cases assume the relationship is linear. b. At the intersection point of the two lines the growth of the population is zero. Why? (Note: We are ignoring all other factors, such as immigration.) The intersection point is called the equilibrium point. At this point the population is said to have stabilized, and the size of the population that corresponds to this point is called the equilibrium number. c. What happens to the equilibrium point if the overall death rate decreases, that is, at each value for population density the death rate is lower? Sketch a graph showing the birth rate and both the original and the changed death rates. Label the graph carefully. Describe the shift in the equilibrium point. d. What happens to the equilibrium point if the overall death rate increases? Analyze as in part (c).

Fines for a particular speeding ticket are defined by the following piecewise function, where \(s\) is the speed in mph and \(F(s)\) is the fine in dollars. $$ F(s)=\left\\{\begin{array}{ll} 0 & \text { if } \quad s \leq 45 \\ 50+5(s-45) & \text { if } 4565 \end{array}\right. $$ a. What is the implied posted speed limit for this situation? b. Create a table of values for the fine, beginning at \(40 \mathrm{mph}\) and incrementing by 5 -mph steps up to \(80 \mathrm{mph}\), and then \(\operatorname{graph} F(s)\) c. Describe in words how a speeding fine is calculated. d. Explain what \(5,10,\) and 20 in the formulas for the respective sections of the piecewise function represent. e. Find \(F(30), F(57),\) and \(F(67)\). f. Graph \(F(s)\).
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