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Chapter 7: Problem 3

\(2 x+y=6 \quad(-2,10),(-1,5),(3,0)\)

Short Answer

Expert verified
Points \((-2, 10)\) and \((3, 0)\) satisfy the equation; \((-1, 5)\) does not.

Step by step solution

01

Understand the Problem

We need to determine if the given points \((-2,10)\), \((-1,5)\), and \((3,0)\) satisfy the equation \(2x + y = 6\). This means substituting the points into the equation to see if the left-hand side equals the right-hand side, which is 6.
02

Substitute First Point

Let's test the point \((-2, 10)\). Substitute \(x = -2\) and \(y = 10\) into the equation:\[ 2(-2) + 10 = -4 + 10 = 6 \]The left side is equal to the right side, so \((-2, 10)\) satisfies the equation.
03

Substitute Second Point

Now let's test the point \((-1, 5)\). Substitute \(x = -1\) and \(y = 5\) into the equation:\[ 2(-1) + 5 = -2 + 5 = 3 \]The left side equals 3, which is not equal to 6, so \((-1, 5)\) does not satisfy the equation.
04

Substitute Third Point

Next, test the point \((3, 0)\). Substitute \(x = 3\) and \(y = 0\) into the equation:\[ 2(3) + 0 = 6 + 0 = 6 \]The left side is equal to the right side, so \((3, 0)\) satisfies the equation.

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Key Concepts

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

Testing Solutions
When given a linear equation, testing a solution involves checking if the coordinates of a point satisfy the equation. To do this, you substitute the x and y values from the point into the equation and see if both sides of the equation are equal. It's like trying a key in a lock; if it turns, the solution fits.

For instance, with the equation \(2x + y = 6\), let's test the point \((-2, 10)\). Substitute \(x = -2\) and \(y = 10\) into the equation, and you get \(2(-2) + 10 = 6\). Breaking it down:
  • Calculate \(2(-2)\) which gives \(-4\).
  • Add 10 to \(-4\), resulting in 6.
    The equation holds true, so \((-2, 10)\) is a valid solution.
Coordinate Points
Coordinate points are pairs of numbers that show positions on a graph. Each point is written as \((x, y)\) where \(x\) is the horizontal position and \(y\) is the vertical. They act like addresses on a map to specify exact spots.

In the context of testing solutions, you're given a set of points, such as \((-2, 10), (-1, 5), (3, 0)\). Each coordinate shows a possible position on the graph for the equation. But not all points will "fit" or satisfy the equation.
  • The point \((-2, 10)\) fits the equation.
  • The point \((-1, 5)\) does not fit the equation. It's like trying a wrong puzzle piece.
  • The point \((3, 0)\) fits perfectly in the equation.
Testing each point helps visualize which points lie on the line represented by the equation.
Equation Substitution
Equation substitution is a method used to test if a coordinate point is a solution to a linear equation. This involves replacing the variables with actual numbers from the points you have been given.

Imagine a simple slot machine: you place values into the slots (or places of x and y). For example, if you have the point \((3, 0)\) and you want to test it against \(2x + y = 6\), you substitute \(x = 3\) and \(y = 0\) into the equation to see if this "pays out" to the value of 6:
  • Substitute \(3\) for \(x\), giving \(2(3)\).
  • Substitute \(0\) for \(y\).
  • Calculate \(2(3) + 0 = 6\).
If both sides equal after substitution, you're confirmed to have a correct match. It鈥檚 an essential step to solving for unknown values and ensuring accuracy in solutions.

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Most popular questions from this chapter

The cost (c) of playing an online computer game for a time \((t)\) in hours is given by the equation \(c=3 t+5\). Label the horizontal axis \(t\) and the vertical axis \(c\), and graph the equation for nonnegative values of \(t\).

\(|3 x+2 y|=6\)

Find \(y\) if the line through \((5,2)\) and \((-3, y)\) has a slope of \(-\frac{7}{8}\)

Find \(x\) if the line through \((x, 4)\) and \((2,-5)\) has a slope of \(-\frac{9}{4}\).

The problem of finding the perpendicular bisector of a line segment presents itself often in the study of analytic geometry. As with any problem of writing the equation of a line, you must determine the slope of the line and a point that the line passes through. A perpendicular bisector passes through the midpoint of the line segment and has a slope that is the negative reciprocal of the slope of the line segment. The problem can be solved as follows: Find the perpendicular bisector of the line segment between the points for the following. Write the equation in standard form. (a) \((-1,2)\) and \((3,0)\) Find the perpendicular bisector of the line segment between the points \((1,-2)\) and \((7,8)\). The midpoint of the line segment is \(\left(\frac{1+7}{2}, \frac{-2+8}{2}\right)\) \(=(4,3)\). The slope of the line segment is \(m=\frac{8-(-2)}{7-1}\) \(=\frac{10}{6}=\frac{5}{3}\). Hence the perpendicular bisector will pass through the point \((4,3)\) and have a slope of \(m=-\frac{3}{5}\). $$ \begin{aligned} y-3 &=-\frac{3}{5}(x-4) \\ 5(y-3) &=-3(x-4) \\ 5 y-15 &=-3 x+12 \\ 3 x+5 y &=27 \end{aligned} $$ Thus the equation of the perpendicular bisector of the line segment between the points \((1,-2)\) and \((7,8)\) is \(3 x+5 y=27\). (b) \((6,-10)\) and \((-4,2)\) (c) \((-7,-3)\) and \((5,9)\) (d) \((0,4)\) and \((12,-4)\)
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