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Chapter 10: Problem 13

Two equations and their graphs are given. Find the inter- section point(s) of the graphs by solving the system. $$\left\\{\begin{array}{r}2 x+y=-1 \\\x-2 y=-8\end{array}\right.$$ CAN'T COPY THE GRAPH

Short Answer

Expert verified
The intersection point is (-2, 3).

Step by step solution

01

Setup the Equations

To find the intersection point of the two lines, we start by considering the system of equations:\[\begin{align*}2x + y &= -1 \x - 2y &= -8\end{align*}\]
02

Solve for One Variable

Choose one of the equations to solve for one of the variables. Let's solve the second equation for \(x\):\[x - 2y = -8\]Rearrange to get:\[x = 2y - 8\]
03

Substitute into the Other Equation

Substitute \(x = 2y - 8\) into the first equation:\[2(2y - 8) + y = -1\]
04

Solve for y

Expand and solve the equation:\[4y - 16 + y = -1\]Combine like terms:\[5y - 16 = -1\]Add 16 to both sides:\[5y = 15\]Divide by 5:\[y = 3\]
05

Substitute Back to Find x

Substitute \(y = 3\) back into \(x = 2y - 8\):\[x = 2(3) - 8\]Calculate to find:\[x = 6 - 8 = -2\]
06

Write the Solution as an Ordered Pair

The solution to the system, which is the intersection point of the graphs, is \((-2, 3)\).

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Key Concepts

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

Solving for Variables
When working with systems of equations, you often need to solve for one variable first. This means isolating a variable in one of the equations so you can use its expression in another equation. Let's break this down a bit.
Imagine you have two equations that share the same variables, like our example:
  • Equation 1: \(2x + y = -1\)
  • Equation 2: \(x - 2y = -8\)
To effectively solve this system, you decide to solve the second equation for \(x\). This involves rearranging the equation to express \(x\) in terms of \(y\), resulting in \(x = 2y - 8\). By doing this, you have isolated \(x\) on one side of the equation. Now, you have an expression for \(x\) that you can substitute into the first equation.
Substitution is a powerful tool for solving systems of equations and helps you simplify complex problems step by step.
Intersection Points
The key goal in solving systems of equations is to find intersection points. An intersection point is where two lines on a graph meet, representing a solution to the system.
In our example:
  • After solving for \(y\) using substitution and simplification, you find \(y = 3\).
  • You then substitute \(y = 3\) back into the expression for \(x\), giving you \(x = -2\).
These values \((-2, 3)\) form the coordinates of the intersection point. This means that if you were to graph both equations, the lines would cross at this exact point.
Finding intersection points is crucial because it shows where solutions exist, which can be incredibly useful in real-world problem-solving scenarios like determining market equilibrium or meeting point on a map.
Graphing Equations
Graphing equations can make understanding systems of equations much easier. Visualizing these equations as lines on a graph helps you see where they intersect.
To graph the given equations:
  • For \(2x + y = -1\), rewrite it in slope-intercept form: \(y = -2x - 1\). This lets us quickly plot the y-intercept (-1) and use the slope (-2) to determine other points.
  • For \(x - 2y = -8\), rearrange to \(y = \frac{1}{2}x + 4\). The y-intercept is 4, and the slope is \(\frac{1}{2}\).
By plotting each of these lines, you'll visualize where they cross, confirming your previous algebraic solution of \((-2, 3)\) as the intersection point.
Graphing is a great way to double-check your work and provide a visual representation of where solutions to systems of equations occur. It makes the process of algebra fun and more exciting by adding a visual dimension.

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Most popular questions from this chapter

(a) If three points lie on a line, what is the area of the "triangle" that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points \(\left(a_{1}, b_{1}\right)\) \(\left(a_{2}, b_{2}\right),\) and \(\left(a_{3}, b_{3}\right)\) are collinear if and only if $$\left|\begin{array}{lll} a_{1} & b_{1} & 1 \\ a_{2} & b_{2} & 1 \\ a_{3} & b_{3} & 1 \end{array}\right|=0$$ (b) Use a determinant to check whether each set of points is collinear. Graph them to verify your answer. (I) \((-6,4),(2,10),(6,13)\) (II) \((-5,10),(2,6),(15,-2)\)

Use a calculator that can perform matrix operations to solve the system, as in Example 7 . $$\left\\{\begin{array}{lr} x+\frac{1}{2} y-\frac{1}{3} z = 4 \\ x-\frac{1}{4} y+\frac{1}{6} z = 7 \\ x+ \quad y- \quad z = -6 \end{array}\right.$$

A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce. $$\begin{array}{|l|c|c|c|} \hline & \text { Type A } & \text { Type B } & \text { Type C } \\ \hline \text { Folic acid (mg) } & 3 & 1 & 3 \\ \text { Choline (mg) } & 4 & 2 & 4 \\ \text { Inositol (mg) } & 3 & 2 & 4 \\ \hline \end{array}$$ (a) Find the inverse of the matrix $$ \left[\begin{array}{lll} 3 & 1 & 3 \\ 4 & 2 & 4 \\ 3 & 2 & 4 \end{array}\right] $$ and use it to solve the remaining parts of this problem. (b) How many ounces of each food should the nutritionist feed his laboratory rats if he wants their daily diet to contain 10 mg of folic acid, 14 mg of choline, and 13 mg of inositol? (c) How much of each food is needed to supply 9 mg of folic acid, 12 mg of choline, and 10 mg of inositol? (d) Will any combination of these foods supply 2 mg of folic acid, 4 mg of choline, and 11 mg of inositol?

Use a calculator that can perform matrix operations to solve the system, as in Example 7 . $$\left\\{\begin{array}{l} 3 x+4 y-z=2 \\ 2 x-3 y+z=-5 \\ 5 x-2 y+2 z=-3 \end{array}\right.$$

A publishing company publishes a total of no more than 100 books every year. At least 20 of these are nonfiction, but the company always publishes at least as much fiction as nonfiction. Find a system of inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each year consistent with these policies. Graph the solution set.
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