Question

complete the proof. given: ( ac = 35 ) and ( ab + bc = ac ) prove: ( x = 10 ) image of a number line with points a, b, c and segments ( 2x - 1 ) (ab) and ( x + 6 ) (bc)

statements	reasons
( ab + bc = ac )
( (2x - 1) + (x + 6) = 35 )
( 3x + 5 = 35 )
( 3x + 5 - 5 = 35 - 5 )
( 3x = 30 )
( 3x div 3 = 30 div 3 )
( x = 10 )

reasons bank
given
subtraction property of equality
substitution property of equality
division property of equality
addition property of equality
simplify
multiplication property of equality

Explanation:

Step1: Match first given statement

Statement: $AC = 35$
Reason: Given

Step2: Match second given statement

Statement: $AB + BC = AC$
Reason: Given

Step3: Substitute segment values

Statement: $(2x - 1) + (x + 6) = 35$
Reason: Substitution Property of Equality

Step4: Combine like terms

Statement: $3x + 5 = 35$
Reason: Simplify

Step5: Subtract 5 from both sides

Statement: $3x + 5 - 5 = 35 - 5$
Reason: Subtraction Property of Equality

Step6: Simplify the equation

Statement: $3x = 30$
Reason: Simplify

Step7: Divide both sides by 3

Statement: $3x \div 3 = 30 \div 3$
Reason: Division Property of Equality

Step8: Simplify to find x

Statement: $x = 10$
Reason: Simplify

Answer:

Statements	Reasons
$AB + BC = AC$	Given
$(2x - 1) + (x + 6) = 35$	Substitution Property of Equality
$3x + 5 = 35$	Simplify
$3x + 5 - 5 = 35 - 5$	Subtraction Property of Equality
$3x = 30$	Simplify
$3x \div 3 = 30 \div 3$	Division Property of Equality
$x = 10$	Simplify