How to solve first-degree equation with two or more numbers to the right of the $=$? [closed]

I can solve some equations of the first degree when it has a right number of the symbol $=$ like this one. For example,

$$2x+2=12$$

But when the equation has two or more numbers I cannot solve it. Look at this equation. For example,

$$23x-16=14-17x$$

I get confused when I try to solve an equation like that. How could I learn solve it?

Please excuse my English.

closed as off-topic by José Carlos Santos, max_zorn, Xander Henderson, Isaac Browne, Rhys SteeleJul 31 '18 at 10:40

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• Just move the constant terms on the right side, linear term on the left side – Ṁữŀlɪgắnậcễơưṩ ᛗ Jul 30 '18 at 21:50
• If you have an equation, you may add the same thing to both sides and this will not change the solution to the equation. For $2x+2=12$ this is seen by "adding $-2$ to both sides" giving $2x+2=12\implies 2x+2+\color{red}{(-2)}=12+\color{red}{(-2)}$ which simplifies to $2x=10$. The same technique is used in the second problem you mention, here adding $16$ to both sides and adding $17x$ to both sides, noting that what you add to both sides does not have to be limited to just constants, but you may add variables as well. – JMoravitz Jul 30 '18 at 21:50

To solve this kind of equations you relay on two "tricks" that let's you go through various equivalent equations all the way to the solution. This two "tricks" are listed here.

Basically they say this (note that LHS stands for Left Hand Side and RHS for Right Hand Side)

1. Adding or subtracting a same quantity to the LHS and RHS of an equation, generates an equivalent equation.

2. Multiplying or dividing by the same quantity (different from zero) the LHS and RHS of an equation, generates an equivalent equation

By equivalent equation I mean an equation that has the same solution as the one you started with. Let's see how this principles are useful to solve your equation (I'll refer them as $(1)$ and $(2)$) \begin{align}&23x-16=14-17x \\ &\overset{(1)}{\longrightarrow} 23x\color{red}{+17x}-16=14-17x\color{red}{+17x} \\ &\overset{(1)}{\longrightarrow} 40x-16 \color{red}{+16}=14\color{red}{+16} \\&\longrightarrow 40x = 30 \\&\overset{(2)}{\longrightarrow}\frac{40x}{\color{blue}{40}} = \frac{30}{\color{blue}{40}} \\&\longrightarrow x = \frac{3}{4}\end{align} so as you can see by only invoking this two principles you can go through many equivalent equations all the way to the simples equation of the form $x=b$ which is the solution to your problem!

You could ask why I choose to add, subtract, divide and multiplay by a specific number and the reason is to be found, to put it simple, in the fact that to solve an equation you want to have all the terms with the $x$ on one side and all the numbers on the other. So for example in the fist step I don't want that $+17x$ factor on the RHS so I try to remove it: what is a better way to remove it if not annihilate it by subtracting the same factor from both sides? By this means you get $+17x-17x=0$ on the RHS as we wanted.

In fact the same first rule that I told you before can be simplified in this manner

1. Moving a term from the RHS to the LHS, and vice versa, in an equation, changes the sing of that term.

Hint: You can move some of the terms on the right to the left. Using your example,

$$23x-16=14-17x\implies 40x-16=14.$$

• Note for OP: This is achieved here by adding $17x$ to both sides of the equation. Provided you do the same to both sides of the equation you can add or subtract any term you like. You can multiply or divide too - but you have to take care not to divide by zero, and multiplying by zero just gives the equation $0=0$. – Mark Bennet Jul 30 '18 at 21:53
• What is the meaning of symbol $\implies$? – gato Jul 30 '18 at 22:19
• $A\implies B$ is read as "A implies B." It means that if $A$ is true, then $B$ is true as well. – Carl Schildkraut Jul 30 '18 at 22:20

Hint:

Add $17x + 16$ to both sides of the equation (to move all quantities containing $x$ to one side and the constants to the other). Then divide by $40$ to get $x$.

We have that

$$23x-16=14-17x$$

$$23x-16\color{red}{+17x+16}=14-17x\color{red}{+17x+16}$$

$$40x=30$$

$$\frac{40x}{\color{red}{40}}=\frac{30}{\color{red}{40}}$$

$$x=\frac34$$

Golden Rule : Jump over the equal sign and change the sign.

Example $1$: $$3x+12=2x+20$$

$$3x-2x=20-12$$

$$x=8$$

Example $2$

$$-3x+12-25=x-4+10$$

$$-3x-x=-4+10-12+25$$

$$-4x=19$$

$$x=-19/4$$