A kosarad üres
Solving equations graphically
Solving an equation is a process in the course of which we determine the roots of the equation. Naturally we also consider the following as a solution: showing that there are no solutions on the designated fundamental set or getting infinitely many roots.
We can use several different methods while solving an equation. Henceforth we review some of these methods, affixing that for every exercise it is at the solver's discretion to decide which way to follow.
The graphical method relates to one of the determinations given for equations:
We consider the two sides of the equation as assignment rules of functions, and after plotting these we look for the intersection points of their graphs. The \latex{x-}coordinate of the common point gives the root of the equation.
Example 1
Let us solve equation \latex{2x - 2 = 7 - x} graphically.
Solution
The fundamental set of the equation on which we are looking for the solutions is \latex{\R}.
The functions corresponding to the two sides the graphs of which we plot:
\latex{f (x) = 2x - 2}; \latex{g(x) = 7 - x}.
Figure 3
\latex{y=2x-2}
\latex{y=7-x}
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ x }
\latex{ -1 }
\latex{ -2 }
\latex{ -1 }
\latex{ -2 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ y }
The two linear functions have only one point in common the x-coordinate of which is 3. Thus the solution of the equation is \latex{x = 3}. (Figure 3)
Checking: \latex{f(3) =2\times 3-2 = 4}; \latex{g(3) = 7-3=4}.
It can be seen that \latex{f (3) = g(3)}, therefore \latex{x = 3} is indeed a solution.
Example 2
Let us solve the following equation: \latex{\mid x+5\mid=-\frac{1}{2}x-1}.
Solution
The fundamental set is the set of real numbers: \latex{\R}.
The functions of the right and the left side:
Figure 4
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ y }
\latex{ 1 }
\latex{ -2 }
\latex{ -4 }
\latex{ -3 }
\latex{ -5 }
\latex{ -6 }
\latex{ -7 }
\latex{ -8 }
\latex{ -1 }
\latex{ x }
\latex{ f }
\latex{ g }
\latex{f(x)=\mid x+5\mid}; \latex{g(x)=-\frac{1}{2}x-1}.
The abscissas of the intersection points: \latex{x_1 = -8;\; x_2 = -4}. (Figure 4)
Checking the resulting solutions:
\latex{f(-4)=\mid (-4)+5\mid=1};
\latex{g(-4)=-\frac{1}{2}\times(-4)-1=1};
\latex{f(-8)=\mid (-8)+5\mid=3};
\latex{g(-8)=-\frac{1}{2}\times(-8)-1=3}.
So we get that \latex{f(-4) = g(-4)} and \latex{f(-8) = g(-8)}.
Example 3
A number is \latex{2} greater than another one. The sum of their absolute values is also \latex{2}. Give these real numbers.
Solution
Let the smaller number be x, then the larger number is \latex{x + 2}. According to the exercise we can set up the following equation: \latex{\mid x\mid+\mid x + 2\mid= 2}.
The fundamental set is: \latex{\R}. The functions corresponding to the right and the left side:
\latex{f (x) = \mid x\mid+\mid x + 2\mid}; \latex{g(x) =2 }.
When plotting the graphs of the functions we can see that function \latex{f} takes the substitution value \latex{2} at every \latex{x} of closed interval \latex{[-2; 0]} (Figure 5).
Thus the equation set up has infinitely many roots, the solution is: \latex{x\in [-2; 0]} or \latex{-2\leq x\leq0.} In this case we cannot check the equation by substitution.
Thus the equation set up has infinitely many roots, the solution is: \latex{x\in [-2; 0]} or \latex{-2\leq x\leq0.} In this case we cannot check the equation by substitution.
⯁ ⯁ ⯁
This solution method is quite representative, but round about, and it often makes possible only inaccurate reading of the solution. Therefore the examination of other solution methods is necessary; please note that the graphical solution can be unavoidable in the case of solving some equation types.
Such a case would be for example while solving equation \latex{x^{2}=\sqrt{x+2}}, which can be solved the algebraic way only with the help of higher level mathematical tools. Now we can find the roots only by using the graphical method.
\latex{y=2}
\latex{y=\mid x \mid+\mid x+2\mid}
Figure 5
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ y }
\latex{ 1 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -4 }
\latex{ x }
Exercises
{{exercise_number}}. Solve the following equations graphically.
- \latex{2x-1=1-2x}
- \latex{\mid x\mid-\mid x+2\mid=1}
- \latex{\mid 3x-2\mid=2x-1}
- \latex{\mid 3x-2\mid=\mid 3x-1\mid-1}
{{exercise_number}}. For which real number is it true that its absolute value is \latex{1} greater than the number?
{{exercise_number}}. Is there a real number the absolute value of which is \latex{1} less than the number?
{{exercise_number}}. For which real number is it true that the negative of its reciprocal is \latex{2} less than the number?
