Solve for X

Judy and his friends were made to stand in rows, for a game. Judy found that if one of them is extra in a row, there would be 2 rows less and If one of them is less in a row there would be 3 rows more.

To figure out this situation he needs to learn the concept known as "Solve for x".

Solve for x

Lesson Plan

What Does Solve for x Mean?

Solve for x means finding the value of \(x\)  for which the equation holds true. i.e when you find the value of x and substitute in the equation, we should get

LHS = RHS

If I ask you to solve the equation '\(x + 1 = 2\)' that would mean finding some value for x that satisfies the equation.

Do you think \(x=1\) is the solution to this equation? Substitute it in the equation and see.

That’s what solving is all about.


How Do You Solve For x?

To solve for x, bring the variable to one side, and bring all the remaining values to the other side by applying arithmetic operations on both sides of the equation.

Simplify the values to find the result.

Let’s start with simple equation as,  \(x + 2 = 7\)

How do you get x by itself?

Subtract 2 from both sides.

\[\begin{align}
 x + 2 - 2 &= 7 - 2 \\[0.2cm]
 x &= 5
 \end{align}\]

Now, check the answer \(x=5\) by substituting it back into the equation. We get 5 + 2= 7. But is this true?

Yes, so the solution (or answer) is \(x=5\) is correct.


Solve For x in The Triangle

"Solve for x" the unknown side or angle in a triangle we can use triangle property or the Pythagorean theorem.

Let us understand solve for x geometry by an example.

Example

 \(\triangle\ ABC\) is right-angled at \(B\) with two of its legs measuring 7 units and  24 units. can you find the hypotenuse \(x\)?

In \(\triangle ABC\) by using the Pythagorean theorem,

we get

\[\begin{align}
 AC^2 &= AB^2 + BC^2 \\[0.2cm]
 x^2 &= 7^2 + 24^2 \\[0.2cm]
 x^2 &= 49 + 576 \\[0.2cm]
 x^2 &= 625 \\[0.2cm]
 x &= \sqrt{625} \\[0.2cm]
x&=25\text{ units} \\
 \end{align}\]

We also use properties of triangles to solve for \(x\) in a triangle.


Solve Equations in Different Variables

 We can use a system of equations solver and solve equations with different variables.

We solve one of the equations for one variable (for example solve for x in terms of y) and then substitute it in the second equation.

We then solve this for the variable.

Finally, we substitute the value of the variable that we found in one of the equations and solve for the other variable.

Let us understand Solve for x and y by one example.

Example

Solve : \(2x - y = 5, 3x + 2y = 11\)

Solution

\[\begin{align}
 2x - y = 5 \\ 
 \text{Adding y on both sides we get}, \\
 2x - y + y = 5 + y \\
 2x = 5 + y \\
 x = \frac{{5 + y}}{2} \\   \end{align}\]

Above equation is known as x in terms of y.

\[\begin{align} \text{Substitute x} = \frac{{5 + y}}{2}\text{in equation 2} \\
 3(\frac{{5 + y}}{2}) + 2y = 11 \\
 \frac{{15 + 3y}}{2} + 2y = 11 \\
 \frac{{15 + 3y + 4y}}{2} = 11 \\
 \frac{{15 + 7y}}{2} = 11 \\
 15 + 7y = 22 \\
 7y = 22 - 15 \\
 7y = 7 \\
 y = 1 \\
 \text{Substitute y = 1 in x =} \frac{{5 + y}}{2} \\
 x = \frac{{5 + 1}}{2} = \frac{6}{2} = 3 \\
 \end{align}\]

Thus, the solution of the given system of equations is: \[x=3\\y=1\]

A system of two equations can be solved using different methods which you can visualize using the following algebra calculator. You can enter any two equations in two variables and choose a method. It will show the step-by-step solution in this simplify calculator.

 
important notes to remember
Important Notes
  • To solve for x (the unknown variable in the equation), add apply arithmetic operations to isolate the variable.
  • For solving 'x' number of equations we need exactly 'x' number of variables.
  • Solve for x and y can be done by the substitution method, elimination method, cross-multiplication method, etc.

Solved Examples

Example 1

 

 

\(\begin{align}
 \text {Solve for x :}\frac{1}{2}(3x + 1) - \frac{1}{3}(5x + 2) &= x - 1 \\
 \end{align}\)

Solution

\[\begin{align}
 \frac{1}{2}(3x + 1) - \frac{1}{3}(5x + 2) &= x - 1 \\[0.2cm]
 \frac{{3x + 1}}{2} - \frac{{5x + 2}}{3} &= x - 1 \\[0.2cm]
 \frac{{3(3x + 1) - 2(5x + 2)}}{6} &= x - 1 \\[0.2cm]
 \frac{{9x + 3 - 10x - 4}}{6} &= x - 1 \\[0.2cm]
 \frac{{ - x - 1}}{6} &= x - 1 \\[0.2cm]
\text{Multiplying both sides by 6,}\\[0.2cm]
  - x - 1 &= 6(x - 1) \\[0.2cm]
  - x - 1 &= 6x - 6 \\[0.2cm]
  - x - 6x &=  - 6 + 1 \\[0.2cm]
  - 7x &=  - 5 \\[0.2cm]
 x &= \frac{{ - 5}}{{ - 7}} \\[0.2cm]
 x &= \frac{5}{7} \\[0.2cm]
 \end{align}\]

\[x = \frac{5}{7}\]
Example 2

 

 

If one side of the chessboard is smaller than its perimeter by 18 inches, then find the side of the chessboard?

Solution

Let the side of chessboard = '\(x\)' inches

Since the chessboard is square, Therefore its perimeter will be '\(4x\)' inches

According to the given condition,

\[\begin{align}
 \text{Perimeter} &= x + 18 \\[0.2cm]
 4x &= x + 18 \\[0.2cm]
 4x - x &= 18 \\[0.2cm]
 3x &= 18 \\[0.2cm]
 x &= \frac{{18}}{3} \\[0.2cm]
 x &= 6 \\
 \end{align}\]

Therefore

\[  \text{The side of the chessboard is 6 inches  } \]
Example 3

 

 

The sum of the digits of a two-digit number is 9. The digits of the number are reversed if 45 is added to it. Find the original number.

Solution

Let the one's place digit be '\(x\)'

And ten's place digit be  '\(y\)'

Then, the original number = \(10y + x\)

The reversed number = \(10x + y\)

According to the given condition,

\[\begin{align}
 x + y = 9.........(1) \\[0.2cm]
\text{Adding 45 to the original number}\\
 10y + x + 45 = 10x + y \\[0.2cm]  \because \text{number gets reversed}\\
 45 = 10x - x + y - 10y \\[0.2cm]
 45 = 9x - 9y \\[0.2cm]
 \text{Dividing throughout by 9 we get,} \\[0.2cm]
 5 = x - y...........(2) \\ 
 \end{align}\]

Adding statement (1) & (2) we get,

\[\begin{align}
 (x + y) + (x - y) &= 9 + 5 \\[0.2cm] 
 2x &= 14 \\[0.2cm] 
 x &= \frac{{14}}{2} \\[0.2cm]
 x &= 7 \\[0.2cm]
 \end{align}\]

Now, substitute \(x=7\) in statement 1 , we get

\(\begin{align}
 x + y &= 9 \\[0.2cm]
 7 + y &= 9 \\[0.2cm]
 y &= 9 - 7 \\[0.2cm]
 y &= 2
 \end{align}\)

Therefore, the number is 27.
Example 4

 

 

The ages of Roony and Herald are in the ratio 5: 7

If four years later, the sum of their ages will be 56 years, then how old is  Roony now?

Solution

The ratio of Rooney and Herald's age is 5:7

Let the Ratio of Rooney's to Herald's age = \(  \dfrac{5}{7} = \dfrac{{5x}}{{7x}}\)

The ratio of their ages after 4 years  will be\( = \dfrac{{5x + 4}}{{7x + 4}}\)

According to given condition,

\[\begin{align}
 (5x + 4) + (7x + 4) &= 56 \\[0.2cm]
 5x + 7x + 4 + 4 &= 56 \\[0.2cm]
 12x + 8 &= 56 \\[0.2cm]
 12x &= 56 - 8 \\[0.2cm]
 12x &= 48 \\[0.2cm]
 x &= \frac{{48}}{{12}} \\[0.2cm]
 x &= 4 \\[0.2cm]
 \end{align}\]

Roony's age is \(5 \times x = 5 \times 4 = 20\)

 Therefore, the present age of Roony is 20 years 
Example 5

 

 

A total of $1380 is divided between Chris, John, and Ashley in such a way that Chris gets 5 times as much as Ashley's share and 3 times as much as John's share. If the amount received by Chris and John is the same, can you find out Ashley's share?

Solution

Let Ashley's share be \(x\) dollars.

Then, Chris's share = $\(5x\)

Therefore, John's share = Total amount -(Ashley's share + Chris's share)

\(\begin{array}{l}
  = [1380 - (x + 5x)] \\[0.2cm]
  = 1380 - 6x \\[0.2cm]
 \end{array}\)

According to the given condition,

\[\begin{align}
 5x &= 3(1380 - 6x) \\[0.2cm]
 5x &= 4140 - 18x \\[0.2cm]
 5x + 18x &= 4140 \\[0.2cm]
 23x &= 4140 \\[0.2cm]
 x &= \frac{{4140}}{{23}} \\[0.2cm]
 x &= 180 \\
 \end{align}\]

Therefore, Ashley's share is $180
 
Challenge your math skills
Challenging Questions
  • Advanced purchase tickets to an exhibition cost $4, while tickets purchased at the door cost $6. If a total of 150 tickets were sold and $680 was collected, how many advanced tickets were sold?
  • If the length of a rectangle is increased and the width is reduced by 3 units each, the area is reduced by 32 square units. However, if the length is decreased by 3 unit and the width is increased by 2 units, the area will increase by 34 square units. Find the rectangle area.

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 
 
 

Let's Summarize

This mini-lesson targeted the fascinating concept of "solve for x". The math journey around solve for x starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. How do you solve x in a bracket?

Use distributive law and remove the bracket, move all the x terms to one side and constant to the other side and find the unknown \(x\).

Example :

\(\begin{align}
 2(x - 3) &= 4 \\[0.2cm]
\text{By using} &\text{ Distributive law}\\ 
 2x - 6 &= 4 \\[0.2cm] 
 2x &= 4 + 6 \\[0.2cm]  
 2x &= 10 \\[0.2cm]  
 x &= \frac{{10}}{2} \\[0.2cm]  
 x &= 5 \\ 
 \end{align}\)

2.  How do you solve x in a fraction?

Eliminate the denominator by cross multiplication and then solve for x.

Example

\(\begin{align}
 \frac{x}{4} + \frac{1}{2} &= \frac{5}{2} \\[0.2cm]
 \frac{{2x + 4}}{8}& = \frac{5}{2} \\[0.2cm] 
\text{By doing cross}& \text{ multiplication we get, }\\[0.2cm]
 2(2x + 4)& = 8(5) \\[0.2cm] 
 4x + 8 &= 40 \\[0.2cm] 
 4x &= 40 - 8 \\[0.2cm] 
 4x &= 32 \\[0.2cm] 
 x &= \frac{{32}}{4} \\[0.2cm] 
 x &= 8
 \end{align}\)

  
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