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Chapter 5: Problem 22

Solve the system by the method of substitution. $$\left\\{\begin{aligned} x-2 y &=-2 \\ 3 x-y &=6 \end{aligned}\right.$$

Short Answer

Expert verified
The solution to the system of equations is \(x = 2\), \(y = 0\).

Step by step solution

01

Rearrange Equation

Rewrite the first equation to isolate 'x'. From the first equation, \(x= 2y + 2\). This is the expression that will be substituted into the second equation.
02

Substitute the Expression into the Second Equation

Substitute the expression obtained in Step 1 into the second equation \[3x - y = 6\], replacing x with \(2y +2\). The equation becomes \[3(2y + 2) - y = 6\].
03

Simplify and Solve for y

First expand to get \[6y + 6 - y = 6\]. Then combine like terms: \(5y + 6 = 6\). Now, subtract 6 from both sides of the equation to isolate y: \(5y =0\), hence \(y=0\).
04

Find the Value of x

Now that we have the value of y, we can substitute it back into the equation from Step 1: \(x =2(0) + 2\). Therefore, \(x = 2\).

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique used to solve systems of equations, which are collections of two or more algebraic equations containing the same variables. Understanding the substitution method allows you to find the intersection point between lines, showing where the equations hold true simultaneously.

To apply the substitution method effectively, follow these steps:
  • Rearrange one of the equations to isolate one variable.
  • Substitute the isolated variable's expression into the other equation.
  • Solve the resulting equation for one variable.
  • Replace the found value back into any of the original equations to get the other variable's value.

By replacing variables with their equivalent expressions, you reduce a system of multiple variables to a single-variable equation, which is usually much easier to solve. When a system's solution exists, the substitution method will lead you to a specific point, \(x, y\), that satisfies both equations.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations that represent a specific quantity. For example, in the system of equations we're considering, \(x=2y+2\) is an algebraic expression representing the value of \(x\) in terms of \(y\).

Here are some important aspects of algebraic expressions:
  • They consist of terms, which are the separated parts of the expression, usually added or subtracted.
  • Each term has a coefficient (the numerical factor) and may have one or more variables raised to a power (the degree).
  • When you work with algebraic expressions, manipulate them using the properties of real numbers, such as associative, commutative, and distributive laws to simplify or rearrange the terms.

Understanding algebraic expressions is crucial for solving equations as they convey the relationship between variables and can be manipulated to isolate specific variables.
Isolate Variables
To isolate variables means to manipulate an equation in such a way that you get a specific variable alone on one side of the equation. Isolating variables is integral to the substitution method because it sets the stage for replacing variables with their equivalent expressions in other equations.

Consider these strategies to isolate a variable effectively:
  • Use basic arithmetic operations (addition, subtraction, multiplication, and division) to modify both sides of the equation and move the desired variable to one side.
  • When dealing with fractions, multiply through by the least common denominator to eliminate them.
  • For equations with the variable in the exponent, apply logarithms to bring down the exponent.
  • If the variable is within a function (like trigonometric, exponential, or logarithmic), apply the inverse function to free it.

Isolating variables may require several steps and a clear understanding of inverse operations. It's a fundamental skill that simplifies complex problems and helps reveal solutions to equations.

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Most popular questions from this chapter

Computers The sales \(y\) (in billions of dollars) for Dell Inc. from 1996 to 2005 can be approximated by the linear model \(y=5.07 t-22.4, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. (Source: Dell Inc.) (a) The total sales during this ten-year period can be approximated by finding the area of the trapezoid represented by the following system. \(\left\\{\begin{array}{l}y \leq 5.07 t-22.4 \\ y \geq 0 \\ t \geq 5.5 \\ t \leq 15.5\end{array}\right.\) Graph this region using a graphing utility. (b) Use the formula for the area of a trapezoid to approximate the total sales.

Concert Ticket Sales Two types of tickets are to be sold for a concert. One type costs $$\$ 20$$ per ticket and the other type costs $$\$ 30$$ per ticket. The promoter of the concert must sell at least 20,000 tickets, including at least 8000 of the $$\$ 20$$ tickets and at least 5000 of the $$\$ 30$$ tickets. Moreover, the gross receipts must total at least $$\$ 480,000$$ in order for the concert to be held. (a) Find a system of inequalities describing the different numbers of tickets that must be sold, and (b) sketch the graph of the system.

Sketch the graph of the inequality. $$y<\ln x$$

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x^{2}+y^{2} \leq 25 \\ x^{2}+y^{2} \geq 9\end{array}\right.$$

Reasoning An objective function has a minimum value at the vertex \((20,0)\). Can you conclude that it also has a minimum value at the point \((0,0)\) ? Explain.
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