Solution of an equation by adding/subtracting two sides - Examples, Exercises and Solutions

Solve the equation and find Y:

$20\times y+8\times2-7=14$

First, we will put parentheses around the two multiplication exercises:

$(20\times y)+(8\times2)-7=14$

We solve the exercises within the parentheses:

$20y+16-7=14$

We simplify:

$20y+9=14$

We move the sections:

$20y=14-9$

$20y=5$

We divide by 20:

$y=\frac{5}{20}$

$y=\frac{5}{5\times4}$

We simplify:

$y=\frac{1}{4}$

$\frac{1}{4}$

$92-x\times2-24\colon4=64$Calculate X.

First, we solve the multiplication and division exercises, we will put them in parentheses to avoid confusion:

$92-(x\times2)-(24\colon4)=64$

$92-2x-6=64$

Reduce:

$86-2x=64$

Move the sides:

$-2x=64-86$

$-2x=-22$

Divide by negative 2:

$x=\frac{-22}{-2}$

$x=11$

11

Solve for X:

$5x-8=10x+22$

First, we arrange the two sections so that the right side contains the values with the coefficient x and the left side the numbers without the x

Let's remember to maintain the plus and minus signs accordingly when we move terms between the sections.

First, we move a$5x$ to the right section and then the 22 to the left side. We obtain the following equation:

$-8-22=10x-5x$

We subtract both sides accordingly and obtain the following equation:

$-30=5x$

We divide both sections by 5 and obtain:

$-6=x$

$-6$

What is the missing number?

$23-12\times(-6)+?\colon7=102$

First, we solve the multiplication exercise:

$12\times(-6)=-72$

Now we get:

$23-(-72)+x\colon7=102$

Let's pay attention to the minus signs, remember that a negative times a negative equals a positive.

We multiply them one by one to be able to open the parentheses:

$23+72+x\colon7=102$

We reduce:

$95+x:7=102$

We move the sections:

$x:7=102-95$

$x:7=7$

$\frac{x}{7}=7$

Multiply by 7:

$x=7\times7=49$

49

Solve for X:

$\frac{1}{8}x-\frac{2}{3}x+\frac{1}{5}x=1$

The common denominator of 8, 3, and 5 is 120.

Now we multiply each numerator by the corresponding number to reach 120 and thus cancel the fractions and obtain the following equation:

$(1\times x\times15)-(2\times x\times40)+(1\times x\times24)=1\times120$

We multiply the exercises in parentheses accordingly:

$15x-80x+24x=120$

We will solve the left side (from left to right) and will obtain:

$(15x-80x)+24x=120$

$-65x+24x=120$

$-41x=120$

We reduce both sides by $-41$

$\frac{-41x}{-41}=\frac{120}{-41}$

We find that x is equal$x=-\frac{120}{41}$

$-\frac{120}{41}$

Fill in the missing number:

$?\colon(-6)+\lbrack-2(94-73)\rbrack=-153$

We solve the innermost parentheses:

$94-73=21$

We obtain:

$x\colon(-6)+\lbrack-2(21)\rbrack=-153$

We will focus on the parentheses and multiply:

$-2\times21=-42$

We obtain:

$x\colon(-6)+(-42)=-153$

Let's consider the multiplication between minus and plus:

$x\colon(-6)-42=-153$

We swap sections:

$x\colon(-6)=-153+42$

$x\colon(-6)=-111$

$\frac{x}{-6}=-111$

We multiply by minus 6:

$x=-111\times-6=666$

$x=666$

Fill in the missing number:

$?\colon4+8\times3-7=26$

First, we place the multiplication exercise in parentheses:

$x\colon4+(8\times3)-7=26$

We solve the multiplication exercise:

$x\colon4+24-7=26$

We place the subtraction exercise in parentheses:

$x\colon4+(24-7)=26$

We solve the subtraction exercise:

$x\colon4+17=26$

We swap sections:

$x\colon4=26-17$

$\frac{x}{4}=9$

Multiply by 4

$x=4\times9=36$

$?=36$

How much is xequal to?

$-25-42\colon x+18\times2\colon4=-23$

First, we will put the multiplication exercise in parentheses:

$-25-42\colon x+(18\times2)\colon4=-23$

$-25-42\colon x+36\colon4=-23$

We will put the division exercise in parentheses:

$-25-42\colon x+(36\colon4)=-23$

$-25-42\colon x+9=-23$

We arrange the exercise so that we can simplify:

$-25+9-42\colon x=-23$

$(-25+9)-42\colon x=-23$

We solve the exercise in parentheses and obtain:

$-16-42\colon x=-23$

We move the fractions and obtain:

$-42\colon x=-23+16$

$-42\colon x=-7$

We multiply by x and obtain:

$-42=-7x$

We divide by negative 7:

$x=\frac{-42}{-7}=6$

6

What is the missing number in the equation?

$5\times17-112-\frac{?}{14}=-27.5$

We place the multiplication exercise in the parenthesis:

$(5\times17)-112-\frac{x}{14}=-27.5$

We solve the multiplication exercise and then subtract:

$85-112-\frac{x}{14}=-27.5$

$-27-\frac{x}{14}=-27.5$

We swap sections:

$-\frac{x}{14}=-27.5+27$

$-\frac{x}{14}=-\frac{1}{2}$

We multiply by 14:

$-x=-\frac{1}{2}\times14$

$-x=-7$

We multiply by -1

$x=7$

$7$

Calculate the missing number in the equation:

$(45-22)\times19+25\colon5+?\times3=82$

We solve the exercise in parentheses:

$23\times19+25\colon5+?\times3=82$

We place in parentheses the multiplication and division exercises:

$(23\times19)+(25\colon5)+?\times3=82$

We solve the exercise in parentheses:

$437+5+?\times3=82$

We simplify:

$442+?\times3=82$

We swap sections:

$?\times3=82-442$

$?\times3=-360$

We divide by 3

$?=-120$

$?=-120$

Daniela goes to the bookshop and buys 4 pens and 9 notebooks for a total of $51.

The price of a pen is twice as much as the price of a notebook.

How much is a pen?

We will identify the price of the notebook with x and since the price of the pen is 2 times greater we will mark the price of the pen with 2x

The resulting equation is 4 times the price of a pen plus 9 times the price of a notebook = 51

Now we replace and obtain the following equation:

\( 4\times2x+9\times x=51

According to the rules of the order of arithmetic operations, multiplication and division operations precede addition and subtraction, therefore we will first solve the two multiplication exercises and then add them up:

$(4\times2x)+(9\times x)=51$

$(4\times2x)=8x$

$(9\times x)=9x$

$8x+9x=17x$

Now the obtained equation is: $17x=51$

We divide both sides by 17 and find x

$x=\frac{51}{17}=3$

As we discovered that x is equal to 3, we will place it accordingly and find out the price of a pen:$2\times x=2\times3=6$

$6$

What is the number that should replace y?

$23-12\times(-5)+y\colon7+\lbrack21-4\rbrack=107$

First, we solve the multiplication exercise:

$12\times(-5)=-60$

and the exercise within brackets:

$21-4=17$

We obtain:

$23-(-60)+y:7+17=107$

Keep in mind that a negative times a negative becomes a positive:

$23+60+y:7+17=107$

We simplify and add:

$23+60=83$

$83+17=100$

We obtain:

$100+y:7=107$

We move the sections:

$y:7=107-100$

$\frac{y}{7}=7$

We multiply by 7:

$y=7\times7=49$

49

Mariana has 3 daughters.

The first daughter is 2 times older than the second daughter.

The second daughter is 5 times older than the third daughter.

If we increase the age of the third daughter by 12 years, she will be the same age as the second daughter.

How old is the the first daughter?

In the first step, we will try to use variables to change the exercise from verbal to algebraic.

Let's start with the third daughter and define her age as X.

The second daughter, as written, is 5 times older than her, so we will define her age as 5X.

The first daughter is 2 times older than the second daughter, so we will define her age as 2*5X, that is, 10X.

Now let's look at the other piece of information: it is known that if we increase the age of the third daughter by 12 years, then she will be the same age as the second sister.

So we will write X+12 (the third daughter plus another 12 years)

=

5X (age of the second daughter)

X + 12 = 5X

Once we have an equation, we can solve it. First, we'll move the sections:

5X-X=12

4X=12

We divide by 4:

X=3

But this is not the solution!

Remember, we were asked for the age of the first daughter, which is 10X

We replace the X we found:

10*3 = 30

This is the solution!

$30$

Solve for X:

$x+3=5$

$2$

Solve for X:

$3-x=1$

$2$