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An equation is an algebraic expression consisting of variables equated to each other using the equal “=” symbol. A linear equation is defined as an equation with a degree of one. Mathematical knowledge is frequently applied through word problems, and linear equations are used to answer these word problems on a large scale.
Understanding how to put up challenges like this one correctly makes finding a solution a simple question of following a pattern. In this part, we’ll answer this and other problems containing three equations and three variables. This is done using strategies similar to those used to solve linear equations. Finding solutions to systems of three equations, on the other hand, needs a little more organisation and a little visual gymnastics.
The main objective of eliminating one variable at a time to achieve back-substitution is to solve systems of equations in three variables, also referred to as three-by-three systems. An ordered triple is a solution to a system of three equations in three variables (x, y, z).
We can carry out the following actions to arrive at a solution:
• Switch up any two equations’ order.
• Multiply an equation’s two sides by a non-zero constant.
• Add one equation to another equation by a nonzero multiple.
Determine whether the ordered triple (3, −2,1) is a solution to the system.
X + Y + Z=2
6x−4y+5z=31
5x+2y+2z=13
Show Solution
We will check each equation by substituting the values of the ordered triple for X,Y,Z.
X + Y+ Z=2 (3) +(−2) +(1) =2 True | 6x−4y+5z=31 6(3) −4(−2) +5(1) =31 18+8+5=31 True | 5x+2y+2z=13 5(3) +2(−2) +2(1) =13 15−4+2=13 True |
The ordered triple (3, −2, 1) is indeed a solution to the system.
Linear Equations
A linear equation is just an algebraic phrase that depicts a line as a reminder. 3x = 9 or y + 4 = 10 are examples of these equations with only one variable. We’re trying to figure out the variable in these equations by getting it alone on one side of the equals sign.
Simple Problems
The writing system of linear equations is used in some simple tasks. The sum of 35 with a number, for example, is 72. What is the figure?
The variable is something we don’t know. Here, let’s use the letter x. We know that x + 35 = 72 is our equation. We get x = 37 by subtracting 35 from both sides of the equation. We now have our phone number.
They can be a little more complicated, such as this: The number 57 is 15 fewer than four times a number. What is the figure? Let’s use x as the number once more. 4x is four times that number. 4x -15 is 15 less than that. So, 4x – 15 = 57 is our equation. To find x, multiply both sides by 15. Then we divide by four, giving us x = 18.
Solve Systems of Three Equations in Three Variables
The basic purpose of writing systems of linear equations is to eliminate one variable at a time to achieve back-substitution. An ordered triple is a solution to a system of three equations in three variables (x,y,z),(x,y,z).
We can use the following operations to obtain a solution:
1. Rearrange any two equations in any order.
2. Add a nonzero constant to both sides of an equation.
3. Substitute a nonzero multiple of one equation for another.
The ordered triple graphic represents the location where three planes in space intersect. Imagine every corner of a rectangular room as a location for such an intersection. Three planes constitute a corner: two opposing walls and the floor (or ceiling). The intersection of three planes is represented by any place where two walls and the floor meet.
There are various problems that include relationships between known and unknown numbers and can be expressed as equations. Because the equations are usually expressed in words, we refer to these issues as word problems. We have already practised equations using one-variable equations to tackle several real-life problems.
Steps involved in solving a linear equation word problem:
Read the problem thoroughly and make a note of what is provided, what is expected, and what is provided.
Using the variables x, y, and so on, denote the unknown.
Translate the problem into mathematical language or mathematical statements.
Using the conditions given in the problems, create a linear equation in one variable.
For the unknown, consider solving the linear equation.
Check to see if the answer meets all of the problem’s requirements.
Solve practical word problems of Linear Equation:
1. The total of two numbers equals twenty-five. One of the numbers is nine times larger than the other. Find the figures.
Solution:
Then the other number = x + 9
Let the number be x.
Sum of two numbers = 25
Coming to the query, x + x + 9 = 25
⇒ 2x + 9 = 25
⇒ 2x is equal to 25 – 9
⇒ 2x is equal to 16
⇒ 2x/2 is equal to 16/2 (divide by 2 on both the sides)
⇒ x is equal to 8
Therefore, x + 9 is equal to 8 + 9 is equal to 17
Therefore, the two numbers are 8 and 17.
Conclusion
When considering the writing system of linear equations, ensure that each step is placed under the previous one, with the equal signs aligned. The logic of the solution flows properly due to this meticulous planning, and the algebra is simple to check. For more difficult equations, unsystematic methods such as guess and check will be of limited use.
