Linear(Simple) Equations: Very Difficult Problems with Solutions

Solution:
The statement is equivalent to the following equation:
[tex]x=7x-18[/tex]
[tex]18=7x-x[/tex]
[tex]6x=18[/tex]
[tex]x=3[/tex]

Solution:
Let us denote the number with n. The problem can be rewritten as [tex]n=4n-75[/tex]. By substracting [tex]n[/tex] from both sides, we get [tex]3n-75=0[/tex]. Now we divide both sides by 3 to get [tex]n-25=0[/tex], or [tex]n=25[/tex].

Solution:
Let us denote the number with n. We rewrite the problem as [tex]2n-12=3n-15[/tex]. We substract [tex]2n[/tex] from both sides and get [tex]-12=n-15[/tex]. Then we add 15 to both sides in order to get [tex]n=3[/tex].

Solution:
[tex]2x=5x+18[/tex]
[tex]3x=-18[/tex]
[tex]x=-6[/tex]

Solution:
(a + b) + (b + c) + (a + c) = 29 + 45 + 44
a + b + b + c + a + c = 118
2a + 2b + 2c = 118
2(a + b + c) = 118
a + b + c = 59

Solution:
Answer: x = -5

Solution:
Multiplying by [tex]12[/tex]:
[tex]12x-3\left(1-\dfrac{3x}{2}\right)=4\left(2-\dfrac{x}{4}\right)-11[/tex]
[tex]12x-3+\dfrac{9x}{2}=8-x-11 \iff 13x+\dfrac{9x}{2}=0 \iff \dfrac{35x}{2}=0 \Rightarrow x=0[/tex]

Solution:
Notice that the first term equals the right side:
[tex]1\frac{1}{4}\left(\frac{x}{5}-\frac{2}{5}\right)=\frac{5}{4}\times \frac{x-2}{5}=\frac{x-2}{4}[/tex]
So the second term must be [tex]0[/tex]:
[tex]1\frac{5}{6}\left(\frac{x}{11}-1\right)=0 \Rightarrow \frac{x}{11}=1 \Rightarrow x=11[/tex]

Solution:
[tex]x=1:\ 3|a|-4+2a=5 \Rightarrow 3|a|+2a=9[/tex]
If [tex]a\ge 0[/tex]: [tex]5a=9[/tex], [tex]a=\frac{9}{5}[/tex] – not an integer.
If [tex]a\lt 0[/tex]: [tex]-3a+2a=9 \Rightarrow a=-9[/tex].
[tex]y=27x-4x-18=23x-18[/tex]
[tex]x=-\frac{1}{3}:\ y=23\times \left(-\frac{1}{3}\right)-18=-\frac{23}{3}-18=-\frac{77}{3}=-25\frac{2}{3}[/tex]