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Chapter 8: Problem 4

Fill in the blank(s). A point of intersection of the graphs of the equations of a system is a _____ of the system.

Short Answer

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Solution

Step by step solution

01

Understanding the term Intersection

The intersection point of the graphs of the equations of a system represents a point at which all the equations in the system hold true. This implies that they have the same x and y values at this point.
02

Identifying the correct term

The term that fills the blank should capture this mutual satisfaction of all equations in the system at a given point. This leads us to the word 'Solution'. A point of intersection represents a solution of the system of equations because all the equations are satisfied by the values at this point.

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Key Concepts

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

Intersection Point
In mathematics, the intersection point can be thought of as a meeting spot for equations. Imagine two friends, each with their own path, and they decide to meet at a specific place. Similarly, when you have two or more equations, their intersection point is where they "meet" on a graph. This represents a set of coordinates, \((x, y)\), where every equation in the system is true.
This point translates into finding a particular solution that works for all equations involved.
For instance, if we have two linear equations such as \(y = 2x + 1\) and \(y = -x + 4\), the intersection is where these lines cross each other on the graph.
  • This point has the same x and y values that satisfy both equations, hence the relation to being a solution.
  • It is crucial because it visually demonstrates where the two relationships described by the equations are equivalent.
Solution of a System
When we speak of the solution of a system of equations, we refer to the values for the variables that satisfy every equation in the system simultaneously. In our context of linear equations, this means finding values of x and y that make both equations true at the same time.
An easy way to understand this is to think of it as a common handshake between the equations. When they all agree, that handshake occurs at the intersection point.
  • The solution can often be found algebraically by techniques such as substitution or elimination, where we work with the equations to isolate these values.
  • Alternatively, as mentioned in our solution exercise, it can be identified graphically by spotting the intersection point on a graph, giving us the visual evidence of what the solution looks like.
Graphing Equations
Graphing equations is like drawing a map for the relationships described by the equations. It's a visual way of understanding mathematical concepts and provides an easier method to locate the solution of a system of equations.
To graph an equation, we typically rewrite it in a form like \(y = mx + b\), where we can easily see the slope \(m\) and y-intercept \(b\). Each equation maps out a specific line on the graph, and where these lines intersect can be immediately observed to identify the solution of the system.
  • Graphing showcases how each equation behaves and intersects in space, helping to clearly illustrate complex relationships in a simpler form.
  • Using graphing, not only can we find solutions, but we can also gain a deeper insight into the nature of the equations themselves, like whether they're parallel or perpendicular.
  • Tools like graphing calculators or graphing software can assist in making this a much faster process and can be particularly useful in checking work done algebraically.

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Most popular questions from this chapter

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l} 4 x-y+z=-5 \\ 2 x+2 y+3 z=10 \\ 5 x-2 y+6 z=1 \end{array}\right.$$

Determine whether the statement is true or false. Justify your answer. Solving a system of equations graphically using a graphing utility always yields an exact solution.

Determine whether the statement is true or false. Justify your answer. If two columns of a square matrix are the same, then the determinant of the matrix will be zero.

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{rr} 7 x-3 y & +2 w=41 \\ -2 x+y & -w=-13 \\ 4 x+z-2 w & =12 \\ -x+y-x & =-8 \end{array}\right.$$

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} x & x \ln x \\ 1 & 1+\ln x \end{array}\right|$$
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