Math 017 Materials With Exercises

Solution:a) To match the form y mx b , subtraction of 4 must be expressed as anaddition. We rewrite y 2x 4 as y 2x ( 4); m 2, b 4.b) To match the form y mx b , we must isolate y on one side of the equation(which means solving the equation for y ). To this end, we add x on both sides and gety x , or equivalently, y 1x 0 ; We see that m 1, b 0 .c) We must solve the equation for y , and then rewrite the right-hand side insuch way that it matches mx b .2y 4x 3 Divide each side by 2. 4x 3y Write the fraction as a sum of two fractions.2 4x3y Simplify.2 23y 2x23As a result, m 2, b .2The definition of a linear equationThe ability to rewrite an expression in a prescribed form is needed to determine if a given equation islinear. We use the following definition.Linear equationin one variableA linear equation in one variable x is an equation that can bewritten in the formax+b = 0,where b is any real number, and a any real number differentfrom zero ( a ≠ 0 ).The equation 2x 3 0 is already in the form ax b 0.2x 3 0ax b 0In this representation a 2,b 3 , and thus 2x 3 0 is an example of a linear equation.To show that the equation 5 x 1 x is also an example of a linear equation, one must be able torewrite it in the prescribed form given in the definition above, namely ax b 0. To this end, wesubtract x from both sides of the equation 5 x 1 x x x , collect like terms, and obtain theequation 4x 1 0 that matches the form ax b 04x1 0ax b 0We can now see that a 4 and b 1, and we have matched to the pattern given in the definition.Therefore, we can conclude that 5 x 1 x is a linear equation in one variable.108