Property 6
The Equality of Two First Degree Equations
\(ax+b=cx+d\), where \(a,b,c,d\) are constants and \(a\neq0,c\neq0\) we have two conditions:
- \(a=c\) (and) \(b=d\Leftrightarrow S.S.=\mathbb R\)
- \(a=c\) (and) \(b\neq d\Leftrightarrow S.S.=\varnothing\)
1.
\(2x-\left\{-3\left(2x+1\right)\right\}=mx+8\)
\(S.S.=\varnothing\)
\(\Rightarrow m=?\)
2.
\(4x+3\left(x+2\right)=mx+9\)
\(S.S.=\varnothing\)
\(\Rightarrow m=?\)
3.
\(3\left(x-2\right)+4x=7x+k\)
\(S.S.=\mathbb R\)
\(\Rightarrow k=?\)
4.
\(-2\left(x-3\right)-3\left(x+1\right)=-5x+k\)
\(S.S.=\mathbb R\)
\(\Rightarrow k=?\)
5.
\(2x+4y=k\)
\(\left(x,y\right)=\left(1,3\right)\)
\(\Rightarrow k=?\)
6.
\(3x-y+k=0\)
\(\left(x,y\right)=\left(-2,1\right)\)
\(\Rightarrow k=?\)
7.
\(3y-2x-k=0\)
\(\left(x,y\right)=\left(1,-3\right)\)
\(\Rightarrow k=?\)
Property 7
Is First Degree Equations with Two Unknowns
\(a_1x+b_1y=c_1\)
\(a_2x+b_2y=c_2\)
we have three possibilities;
- \(\samefrac{a_1}{a_2}=\samefrac{b_1}{b_2}=\samefrac{c_1}{c_2}\)
then the solution set has infinitely many solutions. The graphs are the same line (coincidence of the lines)
- \(\samefrac{a_1}{a_2}=\samefrac{b_1}{b_2}\neq\samefrac{c_1}{c_2}\)
then the solution set has no solution. The graphs are two parallel lines
- \(\samefrac{a_1}{a_2}\neq\samefrac{b_1}{b_2}\)
then the solution set has one solution. The graphs are intersecting at a single point
1.
\(2x+ay=6\)
\(4x-6y=12\)
\(n\left(S.S.\right)=\infty\)
\(\Rightarrow a=?\)
2.
\(2x-4y=6\)
\(x+ky=5\)
\(S.S.=\varnothing\)
\(\Rightarrow k=?\)
3.
\(3x+2y=11\)
\(-6x+ky=4\)
\(S.S.=\varnothing\)
\(\Rightarrow k=?\)
4.
\(a\in\mathbb R^+\)
\(ax+4y=2\)
\(9x+ay=7\)
\(S.S.=\varnothing\)
\(\Rightarrow a=?\)
5.
\(2x+y=6\)
\(4x-ky=12\)
\(n\left(S.S.\right)=\infty\)
\(\Rightarrow k=?\)
6.
\(x-2y=2\)
\(3x+ky=6\)
\(n\left(S.S.\right)=\infty\)
\(\Rightarrow k=?\)