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Chapter 8: Problem 40

Solve each system by the method of your choice. $$ \left\\{\begin{array}{l} {(x-1)^{2}+(y+1)^{2}=5} \\ {2 x-y=3} \end{array}\right. $$

Short Answer

Expert verified
The solutions to the system of equations are (0,-5) and (2,1)

Step by step solution

01

Rearrange the linear equation

Rearrange the linear equation \(2x - y = 3\) to make y the subject. This will give \(y = 2x - 3\)
02

Substitute y into the circle equation

Substitute \(y = 2x - 3\) into the circle equation. This will give the equation \((x-1)^2 + (2x-3+1)^2=5 \) which simplifies to \((x-1)^2 + (2x-2)^2=5\) which also simplifies to \(x^2-2x+1 + 4x^2-8x+4=5\)
03

Solve the Equation

Simplify equation to \(5x^2-10x=0\). Factoring out gives \(5x(x - 2)= 0\). Setting this equal to zero gives the roots x = 0 and x = 2.
04

Solve for Corresponding y Values

Substitute x=0 and x=2 into equation for y to get corresponding y values. When x=0, y=2(-1)-3=-2-3=-5. When x=2, y=2(2)-3=4-3=1.

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Key Concepts

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

Understanding Linear Equations
Linear equations are a fundamental concept in algebra. They typically appear in the form of y = mx + c, where m is the slope of the line, and c is the y-intercept. These equations represent straight lines in a two-dimensional plane.
  • Slope (m): The slope indicates the steepness and direction of the line. A positive slope means the line goes upwards, while a negative slope means it goes downwards.
  • Y-intercept (c): The y-intercept is the point where the line crosses the y-axis. It is the value of y when x is zero.
In our specific problem, the linear equation is 2x - y = 3. To solve the system, we first rearrange this equation to make y the subject:
\[ y = 2x - 3 \]
This makes it easier for us to perform substitution later on. By isolating y, we express it in terms of x, preparing to use it in conjunction with the circle equation.
Understanding Circle Equations
A circle equation is another key concept in coordinate geometry. It describes all the points that form a circle on a plane. The most familiar form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\). Here,
  • \((h, k)\) is the center of the circle.
  • \(r\) is the radius.
For our exercise, the circle equation is \((x - 1)^2 + (y + 1)^2 = 5\).
  • The center of this circle is at \((1, -1)\) and the radius is \(\sqrt{5}\).
This equation means all points \((x, y)\) that satisfy it will lie on the circumference of the circle with these particular center coordinates and radius.
The Substitution Method Explained
The substitution method is a powerful technique for solving systems of equations, especially when dealing with a linear and non-linear equation, such as a circle equation.
Here's how the substitution method works:
  • Step 1: Solve one of the equations for one of the variables. In our exercise, we solved the linear equation for y, obtaining \(y = 2x - 3\).
  • Step 2: Substitute this expression for y into the second equation, the circle equation \((x - 1)^2 + (y + 1)^2 = 5\). This gives us an equation solely in terms of x, making it easier to solve for x.
By substituting \(y = 2x - 3\) into the circle equation, \((2x - 3 + 1)^2\) becomes part of the equation. Simplifying this helps to eventually determine the x-values by solving the resulting quadratic equation.
The power of substitution lies in reducing the complexity to a simpler, one-variable equation, allowing us to solve systems of equations systematically and efficiently.

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Most popular questions from this chapter

Solve each system for \(x\) and \(y,\) expressing either value in terms of a or \(b\), if necessary. Assume that \(a \neq 0\) and \(b \neq 0.\) \(\left\\{\begin{array}{l}{4 a x+b y=3} \\ {6 a x+5 b y=8}\end{array}\right.\)

Exercises \(41-43\) will help you prepare for the material covered in the first section of the next chapter. Solve the system: $$ \left\\{\begin{aligned} x+y+2 z &=19 \\ y+2 z &=13 \\ z &=5 \end{aligned}\right. $$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=5 x+6 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+y \geq 10} \\ {x+2 y \geq 10} \\ {x+y \leq 10} \end{array}\right. \end{aligned} $$

Harsh, mandatory minimum sentences for drug offenses account for more than half the population in U.S. federal prisons. The bar graph shows the number of inmates in federal prisons, in thousands, for drug offenses and all other crimes in 1998 and 2010. (Other crimes include murder, robbery, fraud, burglary, weapons offenses, immigration offenses, racketeering, and perjury.) a. In 1998, there were 60 thousand inmates in federal prisons for drug offenses. For the period shown by the graph, this number increased by approximately 2.8 thousand inmates per year. Write a function that models the number of inmates, y, in thousands, for drug offenses x years after 1998. b. In 1998, there were 44 thousand inmates in federal prisons for all crimes other than drug offenses. For the period shown by the graph, this number increased by approximately 3.8 thousand inmates per year. Write a function that models the number of inmates, y, in thousands, for all crimes other than drug offenses x years after 1998. c. Use the models from parts (a) and (b) to determine in which year the number of federal inmates for drug offenses was the same as the number of federal inmates for all other crimes. How many inmates were there for drug offenses and for all other crimes in that year?

Does \(x^{2}+y=10\) define \(y\) as a function of \(x ?\)
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