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The table above shows two pairs of values for the linear function f. The function can be written in the form $left parenthesis x right parenthesis equals a x plus b$ , where a and b are constants. What is the value of a + b ?

hint Hint:

Hint:
To find the value of a + b, first we find the value of a and b . We are given an equation $f left parenthesis x right parenthesis equals a x plus b$ and the value of x and $f left parenthesis x right parenthesis$ at wo points. We use these two points to find two equations in variables a and b , and find their values. The pair $left parenthesis x comma f left parenthesis x right parenthesis right parenthesis$ satisfies the equation $f left parenthesis x right parenthesis equals a x plus b$ , which means, the equation is true when we put the values of x and $f left parenthesis x right parenthesis$ in the equation.

The correct answer is: 3.25

Given,
when x = 8 , we have $f left parenthesis x right parenthesis equals 12$
when x = 12, we have $f left parenthesis x right parenthesis equals 17$
We use these values in the equation $f left parenthesis x right parenthesis equals a x plus b$
Firstly, we have
$12 equals a.8 plus b$
Writing the above equation in standard form,
$8 a plus b equals 12$
Secondly,
$17 equals a.12 plus b$
Writing the above equation in standard form, we have
$12 a plus b equals 17$
Thus we get two linear equations in two variables;
$8 a plus b equals 12$
$12 a plus b equals 17$
Now, we solve these equations to get the values of and .
Subtracting the first equation from the second, we get
$left parenthesis 12 a minus b right parenthesis minus left parenthesis 8 a minus b right parenthesis equals 17 minus 12$
Simplifying the above equation, we get
$left parenthesis 12 minus 8 right parenthesis a minus b plus b equals 5$
$4 a equals 5$
$a equals 5 over 4$
Putting the value of a in the first equation, we get
$text 8. end text 5 over 4 plus b equals 12$
Simplifying, we get
$10 plus b equals 12$
$b equals 12 minus 10$
b = 2
Thus, we get
$a equals 5 over 4 semicolon b equals 2$
Hence,
$a plus b equals 5 over 4 plus 2 equals 13 over 4 equals 3.25$
Thus, the value of a + b is 3.25

Note:
We can take a different approach to solving the linear equations. The above method is called method of elimination. We may also use the method of substitution; which is finding the values one variable, say a , in terms of b , from the first equation and replacing this value in the second equation to get a linear equation in one variable , b . Then solve it to find b, and use it in the previous equation to find a.

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The table above shows two pairs of values for the linear function f. The function can be written in the form , where a and b are constants. What is the value of a + b ?

The correct answer is: 3.25

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