A solution in math is whereby values are assigned to unknown variables to solve an equation. It is a value or number of values whereby the equation becomes equal when they are substituted for the unknown value. A solution to an equation is often referred to as the root of the equation.
The root of the equation usually applies to polynomial equations but is not limited to them. You can solve mathematical solutions both numerically and symbolically. Numerical solutions to equations are whereby only numbers are used as solutions. Symbolic solutions to mathematical equations occur when expressions are used to represent the solutions.
How to write a solution in math
Once you have a proper solution in math, you need to figure out how to express it clearly. You may have to communicate it for a journal, an exam, competition, or work, and it needs to be written down and understood. You may have found a brilliant solution in math, but not expressing it makes it worthless.
It’s helpful when you can prove a mathematical problem as clearly as possible without making an experienced reader or mathematician have to strain too much to understand what you’re trying to express.
Have a plan
As mentioned earlier, it’s essential to have a clear solution in math such that the reader can interpret it without overthinking. What you are trying to express should be clear and concise such that the reader doesn’t doubt your claim.
Have a plan as the first step in writing your solutions. Outline the solution such, as what you need to define, and make sure you write up the different parts of the solution in a specific order. Having an outline will help you write down steps that a reader can easily follow and as make sure you don’t skip any of the essential parts of the equation.
Make it easy to interpret.
A reader should make out the structure of the solution in math before they get to the answer. If the reader finds it difficult to decipher the solution, they get lost while interpreting the formula. Follow the rules when writing a solution in math so that you don’t lose the overall objective. Let’s see some easy steps to follow below to make solutions easier to interpret:
- Write the solution to an equation horizontally and clearly. If the wordings can’t fit within the line, don’t jam them upwards or downwards. Always start a new line if the space is finished.
- Don’t write formulas in cursive as they may be misinterpreted.
- If possible, use blank paper. Using lined paper or graph paper may make the solutions slightly harder to read than writing them down on blank paper.
- If you’re using blank paper, write the formula within the margins. Draw margins on all four sides of the page and make them at least 0.5 inches.
- If using a pencil, try not to erase anything so that you don’t have messy smudges on the page caused by an eraser.
- If you make a mistake, cancel it by drawing a single line through the mistake. If the mistake is a large block, draw an X through it and carry on with the formula. Avoid scribbling out the mistakes.
- If it’s a comprehensive solution to an equation, number the pages so that the reader knows that the same equation has continued onto the next page. Indicate that the continuation is at the bottom of the page to let your audience know that the solution in math has not ended. If some pages are missing, writing ‘Continued’ will help the reader know that the solution continues.
- If you’d like to add something extra at the end that you feel was left out of the formula, add a symbol like an asterisk where you’d like to add the new text. After adding the asterisk, leave a small note, for example, ‘See the proof below.’ you can then write the proof or whatever information you want to add. Avoid using arrows to direct the reader to the additional information.
- Label the equations so that you can easily refer to them at a later time.
- Avoid using too much algebra in the same paragraph. Each notable equation or step should have its line, avoiding cramming too much work in one space.
Always start the formula from the beginning and take it step by step. It lets the reader understand your train of thought.
A solution to an equation should justify every step that leads to the correct answer. A reader should understand how you got to each step and not doubt whether the solution is true or not as they go along.
Name the values
When writing a solution in math, it should be like telling a story in an understandable way that can engage the reader and help them understand it. A well-written solution to an equation should have well-named values to help distinguish them from each other and help find the solutions. Name every character simply, such as x or n, to let the reader know how to use the values.
When writing a solution in math, include a diagram where possible. Drawing a diagram grabs the reader’s attention and makes the solution easier to understand. It would help if you drew it precisely using a compass and a ruler. If you’re using a computer, use a geometry rendering program. Diagrams like the below Venn Diagram jump out of the page and capture the attention:
Let’s have a look at some guidelines that will make any solution in math more clearer:
- If you’re using a theorem that has a name, you don’t need to prove the theorem. Citing the theorem is enough, for example, Pythagoras Theory AC=3.
- If confident that the step you’re using is famous, but you’re not sure what the step is called, you can write down ‘By a well-known theorem, the area of ABC equals rs, with r being the inradius and s the semi perimeter.
- If you’re not sure whether a step is well-known, but you can prove it quickly and in a few words, go ahead. However, if proving the theorem is a lot of work, you can either outline the proof or go more in-depth to prove the solution to the equation. When writing an academic paper and you want to prove a solution to an equation, make sure you do so as it is paramount for academic papers to have proven facts.
- Each step in a string of algebraic steps should follow a good flow stemming from the one before it.
Follow the guidelines above, and you’ll find that getting solutions to those complex equations becomes easier with time.
What is a solution in math example?
Here’s a good example below:
If we let a, b, and c depict the measurement of a triangle’s sides, prove:
a2(a squared)(−a+b+c)+b2(b squared)(a−b+c)+c2(c squared)(a+b−c)≤3abc.
The triangle’s inequality shows that the sum of two sides of a triangle is as great as the length of the third side. The inequalities then exist
a-b-c ≤ 0
b-c-a ≤ 0
c-a-b ≤ 0
Multiplying the above by perfect squares leave each nonpositive on the left. That being said, please note the below equations and inequalities:
(b−c)2(squared)(a−b−c) ≤ 0
(c−a)2(squared)(b−c−a) ≤ 0
(a−b)2(squared)(c−a−b) ≤ 0
or the below:
(b2(squared)−2b(squared)c+c2(squared))(a−b−c) ≤ 0
(c2(squared)−2ac+a2(squared))(b−c−a) ≤ 0
(a2(squared)−2ab+b2(squared))(c−a−b) ≤ 0
Upon adding these equations and inequalities, we get the below:
a2(squared)(a+b−c+a−b+c)+b2(squared)(a+b−c−a+b+c)+c2(squared)(−a+b+c+a−b+c −2a2(squared)b−2ab2(squared)+2abc+2abc−2b2(squared)c−2bc2(squared)−2a2(squared)c+2abc−2ac2(squared)=a2(squared)(2a−2b−2c)+b2(squared)(−2a+2b−2c)+c2(squared)(−2a−2b+2c)+6abc ≤ 0
The result is when we add 6abc to both sides, then divide by 2 we get the below solution in math:
*Please note that where I’ve added the word (squared) that appears in the formulas means that it’s the initial value squared.
Simpler examples of solutions in math are as below:
2y X 1= x. Solve for s
2 X y X 1 = X
X = 2y
2y X 2 = x Solve for x
2 X y X 2 = x
x = 4y
3 X 2x = x Solve for x
3 X 2x = x
x = 6x
What is the solution to the equation?
The solution of the equation is a set of values that make an equation true when substituted for unknowns. It is a number that is put in place of a variable to solve the equation.
What are the three types of solutions in math?
There are three types of solutions to linear equations in math:
- The unique solution
The unique solution states that only one point exists, substituting which L.H.S and R.H.S of an equation become equal. Below is a diagrammatic representation of the linear equation in one variable having a unique solution in math:
- No solution
The linear solution states that there is no solution in math where there is no point at which lines intersect, or the linear equation’s graphs are parallel. Please see the diagrammatic representation below.
- Infinite many solutions
These equations have many solutions when there’s a solution set of infinite points whereby L.H.S and R.H.S of an equation are equal or the straight lines in the graph overlap. Please see the diagrammatic representation below:
What is no solution in math?
No solution in math means that no answer exists to the equation. Every possible value assigned to the variable in the equation still does not make the equation true.
Get step-by-step solutions to your math problems.
Math has, since the beginning of time, been a difficult task for many people. The various functions of addition, subtraction, multiplication, divisions, and fractions all look like foreign symbols to many, whether it’s in school, at work, competitions, or just your run-of-the-mill day-to-day tasks. However, it helps when a problem is broken down into smaller steps. Upon breaking them down, the math problem becomes more manageable.
Use the following steps to solve your math problems and see how easy it gets to find solutions:
- Scrutinize the problem, putting down numbers written in word form such that there are no words in the problem itself.
- Mark any relevant words that explain the goal, for example, sum or less. These may be essential keywords that you may miss when solving the problem in numeric form.
- Draw diagrams if you need to visualize the problem to understand it better. An example will be to draw a triangle if you’re required to solve the triangle’s surface area and have the length and width of the sides.
- Ensure all the units of measurements are the same. For example, kilometers and miles should all be in kilometers.
- Break down the problem into smaller problems and solve them one by one. Start from left to right, solving those numbers in parenthesis first, then multiplication, division, additions, and finally subtractions.
- Double-check your work to see if the units of measurement were used correctly and that the solution makes sense.
In conclusion, math has always been a subject that’s both feared and hated in equal measure. However, we all need to be adept at math at some level because we use it in a wide range of applications as we go about our daily lives. Reach out to Galaxygrades.com to help you reach your full potential in math because they’ve got a lot of information and learning material that will help you. Click order right away!