In mathematics, functions play a vital role in describing and modeling various phenomena. A function is a rule that assigns a unique output value to each input value. The set of input values is called the domain, and the set of output values is called the range. However, in some cases, the range is not the same as the codomain, which can lead to confusion. In this essay, we will explore the difference between codomain and range. Codomain of a Function The “codomain” of a function or relation is a set of values that might possibly come out of it. ... Read More

In geometry, a quadrilateral is a polygon with four sides, while a parallelogram is a special type of quadrilateral with some specific properties that differentiate it from other quadrilaterals. What is Parallelogram? A parallelogram is a specific type of quadrilateral that has two pairs of parallel sides. In other words, the opposite sides of a parallelogram are parallel and congruent, meaning they have the same length. Parallelograms also have opposite angles that are congruent, meaning they have the same measure. Properties of Parallelogram One of the most important properties of a parallelogram is that the opposite sides are parallel. This ... Read More

Exponential growth and exponential decay are two fundamental concepts in mathematics and science that describe how a quantity changes over time. Both exponential growth and decay follow a mathematical model that is characterized by a constant rate of change, but they are fundamentally different in terms of how the quantity changes over time. What is Exponential Growth? Exponential growth refers to a situation where the quantity of interest increases at an exponential rate over time. This means that the rate of increase of the quantity is proportional to the current value of the quantity. In other words, the more the ... Read More

The study of geometry is enjoyable. Sizes, distances, and angles are the primary focus of this branch of mathematics known as geometry. Shapes are the focus of geometry, a branch of mathematics. It's not hard to understand how geometry may be used to solve problems in the actual world. It finds application in a wide range of fields, including engineering, architecture, the arts, sports, and more. Today, we'll talk about a special topic in triangle geometry called congruence. But first, let's define congruence so we may use it. Whenever one figure can be superimposed over the other in such a ... Read More

Given:The given equations are:(i) $\frac{7x-2}{5x-1}=\frac{7x+3}{5x+4}$(ii) $(\frac{x+1}{x+2})^2=\frac{x+2}{x+4}$To do:We have to solve the given equations and verify the answers.Solution:To verify the answer we have to find the values of the variables and substitute them in the equation. Find the value of LHS and the value of RHS and check whether both are equal.(i) The given equation is $\frac{7x-2}{5x-1}=\frac{7x+3}{5x+4}$$\frac{7x-2}{5x-1}=\frac{7x+3}{5x+4}$On cross multiplication, we get, $(5x+4)(7x-2)=(7x+3)(5x-1)$$5x(7x-2)+4(7x-2)=7x(5x-1)+3(5x-1)$$35x^2-10x+28x-8=35x^2-7x+15x-3$On rearranging, we get, $35x^2-35x^2+18x-8x=-3+8$$10x=5$$x=\frac{5}{10}$$x=\frac{1}{2}$Verification:LHS $=\frac{7x-2}{5x-1}$$=\frac{7(\frac{1}{2})-2}{5(\frac{1}{2})-1}$$=\frac{\frac{7}{2}-2}{\frac{5}{2}-1}$$=\frac{\frac{7-2\times2}{2}}{\frac{5-2\times1}{2}}$$=\frac{\frac{7-4}{2}}{\frac{5-2}{2}}$$=\frac{\frac{3}{2}}{\frac{3}{2}}$$=\frac{3}{2}\times\frac{2}{3}$$=1$RHS $=\frac{7x+3}{5x+4}$$=\frac{7(\frac{1}{2})+3}{5(\frac{1}{2})+4}$$=\frac{\frac{7}{2}+3}{\frac{5}{2}+4}$$=\frac{\frac{7+2\times3}{2}}{\frac{5+2\times4}{2}}$$=\frac{\frac{7+6}{2}}{\frac{5+8}{2}}$$=\frac{\frac{13}{2}}{\frac{13}{2}}$$=\frac{13}{2}\times\frac{2}{13}$$=1$LHS $=$ RHSHence verified.(ii) The given equation is $(\frac{x+1}{x+2})^2=\frac{x+2}{x+4}$$(\frac{x+1}{x+2})^2=\frac{x+2}{x+4}$On cross multiplication, we get, $(x+1)^2(x+4)=(x+2)^2(x+2)$$(x^2+2(x)(1)+1^2)(x+4)=(x^2+2(x)(2)+2^2)(x+2)$$x(x^2+2x+1)+4(x^2+2x+1)=x(x^2+4x+4)+2(x^2+4x+4)$$x^3+2x^2+x+4x^2+8x+4=x^3+4x^2+4x+2x^2+8x+8$On rearranging, we get, $x^3-x^3+6x^2-6x^2+9x-12x=8-4$$-3x=4$$x=\frac{-4}{3}$Verification:LHS $=(\frac{x+1}{x+2})^2$$=(\frac{\frac{-4}{3}+1}{\frac{-4}{3}+2})^2$$=(\frac{\frac{-4+3\times1}{3}}{\frac{-4+2\times3}{3}})^2$$=(\frac{\frac{-4+3}{3}}{\frac{-4+6}{3}})^2$$=(\frac{\frac{-1}{3}}{\frac{2}{3}})^2$$=(\frac{-1}{3})^2\times(\frac{3}{2})^2$$=\frac{1}{9}\times\frac{9}{4}$$=\frac{1}{4}$RHS $=\frac{x+2}{x+4}$$=\frac{\frac{-4}{3}+2}{\frac{-4}{3}+4}$$=\frac{\frac{-4+2\times3}{3}}{\frac{-4+4\times3}{3}}$$=\frac{\frac{-4+6}{3}}{\frac{-4+12}{3}}$$=\frac{\frac{2}{3}}{\frac{8}{3}}$$=\frac{2}{3}\times\frac{3}{8}$$=\frac{1}{1}\times\frac{1}{4}$$=\frac{1}{4}$LHS $=$ RHSHence verified.Read More

Given:The given equations are:(i) $\frac{2}{3x}-\frac{3}{2x}=\frac{1}{12}$(ii) $\frac{3x+5}{4x+2}=\frac{3x+4}{4x+7}$To do:We have to solve the given equations and verify the answers.Solution:To verify the answer we have to find the values of the variables and substitute them in the equation. Find the value of LHS and the value of RHS and check whether both are equal.(i) The given equation is $\frac{2}{3x}-\frac{3}{2x}=\frac{1}{12}$$\frac{2}{3x}-\frac{3}{2x}=\frac{1}{12}$LCM of denominators $3x$ and $2x$ is $6x$$\frac{2(2)-3(3)}{6x}=\frac{1}{12}$$\frac{4-9}{6x}=\frac{1}{12}$$\frac{-5}{6x}=\frac{1}{12}$On cross multiplication, we get, $12(-5)=(1)(6x)$$-60=6x$$6x=-60$$x=\frac{-60}{6}$$x=-10$Verification:LHS $=\frac{2}{3x}-\frac{3}{2x}$$=\frac{2}{3(-10)}-\frac{3}{2(-10)}$$=\frac{2}{-30}-\frac{3}{-20}$$=\frac{-1}{15}+\frac{3}{20}$$=\frac{-1\times4+3\times3}{60}$                  (LCM of $15$ and $20$ is $60$)$=\frac{-4+9}{60}$$=\frac{5}{60}$$=\frac{1}{12}$RHS $=\frac{1}{12}$LHS $=$ RHSHence verified.(ii) The given equation is $\frac{3x+5}{4x+2}=\frac{3x+4}{4x+7}$$\frac{3x+5}{4x+2}=\frac{3x+4}{4x+7}$On cross multiplication, we get, $(3x+5)(4x+7)=(3x+4)(4x+2)$$3x(4x+7)+5(4x+7)=3x(4x+2)+4(4x+2)$$12x^2+21x+20x+35=12x^2+6x+16x+8$On rearranging, we get, $12x^2-12x^2+41x-22x=8-35$$19x=-27$$x=\frac{-27}{19}$Verification:LHS $=\frac{3x+5}{4x+2}$$=\frac{3(\frac{-27}{19})+5}{4(\frac{-27}{19})+2}$$=\frac{\frac{3\times(-27)}{19}+5}{\frac{4\times(-27)}{19}+2}$$=\frac{\frac{-81+5\times19}{19}}{\frac{-108+2\times19}{19}}$$=\frac{\frac{-81+95}{19}}{\frac{-108+38}{19}}$$=\frac{\frac{14}{19}}{\frac{-70}{19}}$$=\frac{14}{19}\times\frac{19}{-70}$$=\frac{1}{1}\times\frac{1}{-5}$$=\frac{-1}{5}$RHS $=\frac{3x+4}{4x+7}$$=\frac{3(\frac{-27}{19})+4}{4(\frac{-27}{19})+7}$$=\frac{\frac{3\times(-27)}{19})+4}{\frac{4\times(-27)}{19})+7}$$=\frac{\frac{-81+19\times4}{19}}{\frac{-108+19\times7}{19}}$$=\frac{\frac{-81+76}{19}}{\frac{-108+133}{19}}$$=\frac{\frac{-5}{19}}{\frac{25}{19}}$$=\frac{-5}{19}\times\frac{19}{25}$$=\frac{-1}{1}\times{1}{5}$$=\frac{-1}{5}$LHS ... Read More

Given:The given equations are:(i) $\frac{y-(7-8y)}{9y-(3+4y)}=\frac{2}{3}$(ii) $\frac{6}{2x-(3-4x)}=\frac{2}{3}$To do:We have to solve the given equations and verify the answers.Solution:To verify the answer we have to find the values of the variables and substitute them in the equation. Find the value of LHS and the value of RHS and check whether both are equal.(i) The given equation is $\frac{y-(7-8y)}{9y-(3+4y)}=\frac{2}{3}$$\frac{y-7+8y}{9y-3-4y}=\frac{2}{3}$$\frac{9y-7}{5y-3}=\frac{2}{3}$On cross multiplication, we get, $3(9y-7)=(2)(5y-3)$$3(9y)-3(7)=2(5y)-2(3)$$27y-21=10y-6$On rearranging, we get, $27y-10y=-6+21$$17y=15$$y=\frac{15}{17}$Verification:LHS $=\frac{y-(7-8y)}{9y-(3+4y)}$$=\frac{\frac{15}{17}-(7-8(\frac{15}{17}))}{9(\frac{15}{17})-(3+4(\frac{15}{17}))}$$=\frac{\frac{15}{17}-(7-(\frac{-8\times15}{17}))}{\frac{9\times15}{17}-(3+\frac{4\times15}{17})}$$=\frac{\frac{15}{17}-(7-(\frac{120}{17}))}{\frac{135}{17}-(3+\frac{60}{17})}$$=\frac{\frac{15}{17}-(\frac{7\times17-120}{17})}{\frac{135}{17}-(\frac{3\times17+60}{17})}$$=\frac{\frac{15}{17}-(\frac{119-120}{17})}{\frac{135}{17}-(\frac{51+60}{17})}$$=\frac{\frac{15}{17}-(\frac{-1}{17})}{\frac{135}{17}-(\frac{111}{17})}$$=\frac{\frac{15+1}{17}}{\frac{135-111}{17}}$$=\frac{\frac{16}{17}}{\frac{24}{17}}$$=\frac{16}{17}\times\frac{17}{24}$$=\frac{2}{3}$RHS $=\frac{2}{3}$LHS $=$ RHSHence verified.(ii) The given equation is $\frac{6}{2x-(3-4x)}=\frac{2}{3}$$\frac{6}{2x-(3-4x)}=\frac{2}{3}$$\frac{6}{2x-3+4x}=\frac{2}{3}$$\frac{6}{6x-3}=\frac{2}{3}$On cross multiplication, we get, $3(6)=2(6x-3)$$18=2(6x)-2(3)$$18=12x-6$On rearranging, we get, $12x=18+6$$12x=24$$x=\frac{24}{12}$$x=2$Verification:LHS $=\frac{6}{2x-(3-4x)}$$=\frac{6}{2(2)-(3-4(2))}$$=\frac{6}{4-(3-8)}$$=\frac{6}{4+5}$$=\frac{6}{9}$$=\frac{2}{3}$RHS $=\frac{2}{3}$LHS $=$ RHSHence verified.Read More

Given:The given equations are:(i) $\frac{1-9y}{19-3y}=\frac{5}{8}$(ii) $\frac{2x}{3x+1}=1$To do:We have to solve the given equations and verify the answers.Solution:To verify the answer we have to find the values of the variables and substitute them in the equation. Find the value of LHS and the value of RHS and check whether both are equal.(i) The given equation is $\frac{1-9y}{19-3y}=\frac{5}{8}$$\frac{1-9y}{19-3y}=\frac{5}{8}$On cross multiplication, we get, $8(1-9y)=(5)(19-3y)$$8(1)-8(9y)=5(19)-5(3y)$$8-72y=95-15y$On rearranging, we get, $72y-15y=8-95$$57y=-87$$y=\frac{-87}{57}$$y=\frac{-29}{19}$Verification:LHS $=\frac{1-9y}{19-3y}$$=\frac{1-9(\frac{-29}{19})}{19-3(\frac{-29}{19})}$$=\frac{1+\frac{29\times9}{19}}{19+\frac{3\times29}{19}}$$=\frac{1+\frac{261}{19}}{19+\frac{87}{19}}$$=\frac{\frac{19\times1+261}{19}}{\frac{19\times19+87}{19}}$$=\frac{\frac{19+261}{19}}{\frac{361+87}{19}}$$=\frac{\frac{280}{19}}{\frac{448}{19}}$$=\frac{280}{19}\times\frac{19}{448}$$=\frac{280}{448}$$=\frac{5}{8}$RHS $=\frac{5}{8}$LHS $=$ RHSHence verified.(ii) The given equation is $\frac{2x}{3x+1}=1$$\frac{2x}{3x+1}=1$On cross multiplication, we get, $2x=1(3x+1)$$2x=3x+1$On rearranging, we get, $3x-2x=-1$$x=-1$Verification:LHS $=\frac{2x}{3x+1}$$=\frac{2(-1)}{3(-1)+1}$$=\frac{-2}{-3+1}$$=\frac{-2}{-2}$$=1$RHS $=1$LHS $=$ RHSHence verified.Read More

Given:The given equations are:(i) $\frac{2y+5}{y+4}=1$(ii) $\frac{2x+1}{3x-2}=\frac{5}{9}$To do:We have to solve the given equations and verify the answers.Solution:To verify the answer we have to find the values of the variables and substitute them in the equation. Find the value of LHS and the value of RHS and check whether both are equal.(i) The given equation is $\frac{2y+5}{y+4}=1$$\frac{2y+5}{y+4}=1$On cross multiplication, we get, $2y+5=(1)(y+4)$$2y+5=y+4$On rearranging, we get, $2y-y=4-5$$y=-1$Verification:LHS $=\frac{2y+5}{y+4}$$=\frac{2(-1)+5}{(-1)+4}$$=\frac{-2+5}{-1+4}$$=\frac{3}{3}$$=1$RHS $=1$LHS $=$ RHSHence verified.(ii) The given equation is $\frac{2x+1}{3x-2}=\frac{5}{9}$$\frac{2x+1}{3x-2}=\frac{5}{9}$On cross multiplication, we get, $9(2x+1)=5(3x-2)$$9(2x)+9(1)=5(3x)-5(2)$$18x+9=15x-10$On rearranging, we get, $18x-15x=-10-9$$3x=-19$$x=\frac{-19}{3}$Verification:LHS $=\frac{2x+1}{3x-2}$$=\frac{2(\frac{-19}{3})+1}{3(\frac{-19}{3})-2}$$=\frac{\frac{2\times(-19)}{3}+1}{-19-2}$$=\frac{\frac{-38+1\times3}{3}}{-21}$$=\frac{\frac{-38+3}{3}}{-21}$$=\frac{\frac{-35}{3}}{-21}$$=\frac{-35}{3\times-21}$$=\frac{5}{3\times3}$$=\frac{5}{9}$RHS $=\frac{5}{9}$LHS $=$ RHSHence verified.Read More

Given:The given equations are:(i) $\frac{5x-7}{3x}=2$(ii) $\frac{3x+5}{2x+7}=4$To do:We have to solve the given equations and verify the answers.Solution:To verify the answer we have to find the values of the variables and substitute them in the equation. Find the value of LHS and the value of RHS and check whether both are equal.(i) The given equation is $\frac{5x-7}{3x}=2$$\frac{5x-7}{3x}=2$On cross multiplication, we get, $5x-7=3x(2)$$5x-7=6x$On rearranging, we get, $6x-5x=-7$$x=-7$Verification:LHS $=\frac{5x-7}{3x}$$=\frac{5(-7)-7}{3(-7)}$$=\frac{-35-7}{-21}$$=\frac{-42}{-21}$$=2$RHS $=2$LHS $=$ RHSHence verified.(ii) The given equation is $\frac{3x+5}{2x+7}=4$.$\frac{3x+5}{2x+7}=4$On cross multiplication, we get, $3x+5=4(2x+7)$$3x+5=4(2x)+4(7)$$3x+5=8x+28$On rearranging, we get, $8x-3x=5-28$$5x=-23$$x=\frac{-23}{5}$Verification:LHS $=\frac{3x+5}{2x+7}$$=\frac{3(\frac{-23}{5})+5}{2(\frac{-23}{5})+7}$$=\frac{\frac{-69}{5}+5}{\frac{-46}{5}+7}$$=\frac{\frac{-69+5\times5}{5}}{\frac{-46+5\times7}{5}}$$=\frac{\frac{-69+25}{5}}{\frac{-46+35}{5}}$$=\frac{\frac{-44}{5}}{\frac{-11}{5}}$$=\frac{-44}{5}\times\frac{5}{-11}$$=\frac{4}{1}\times\frac{1}{1}$$=4$RHS $=4$LHS $=$ RHSHence verified.Read More