One number is 4 less than 3 times a second number. If 3 more than two times the first number is decreased by 2 times the second number, the result is 11. Use the substitution method. What is the first number?

Question

Eli Assouline · Accepted Answer

#n_1=8#
#n_2=4# One number is 4 less than #-> n_1=?-4# 3 times #" ........................."->n_1=3?-4# the second number #color(brown)(".........."->n_1=3n_2-4)# #color(white)(2/2)# If 3 more #" ..........................................."-> ?+3# than two times the first number#" ............"->2n_1+3# is decreased by #"..................................."->2n_1+3-?# 2 times the second number#" ................."->2n_1+3-2n_2# the result is 11#color(brown)(" ....................................."->2n_1+3-2n_2=11)# ' ~ ~~~~\~~~~~ #color(white)(.)n_1=3n_2-4# .....................Equation (1) #2n_1+3-2n_2=11#...........Equation (2) Substitute for #n_1# in equation (2) using equation (1) #color(brown)(2n_1+3-2n_2=11)" "color(blue)( ->2(3n_2-4)+3-2n_2=11)# dividing the brackets by #6n_2-8+3-2n_2=11# #4n_2-5=11# To both sides, add 5. #4n_2=16# Split each side in half. #" "color(green)(n_2=4)# ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Substitute #n_2=4# into equation (1) #n_1=3(4)-4# #n_1=12-4# #" "color(green)(n_1=8)# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Check: One number is 4 less than 3 times the second number #8=3(4)-4=8# confirmed!

Hazel Anglin · Answer

undefined Let the first number be $ x $ and the second number be $ y $.

Given: 
1. $ x = 3y - 4 $
2. $ 2x + 3 = 2y + 11 $

Using the substitution method:

Substitute the expression for $ x $ from the first equation into the second equation:

$$ 2(3y - 4) + 3 = 2y + 11 $$

$$ 6y - 8 + 3 = 2y + 11 $$

$$ 6y - 5 = 2y + 11 $$

Subtract $ 2y $ from both sides:

$$ 6y - 2y - 5 = 11 $$

$$ 4y - 5 = 11 $$

Add 5 to both sides:

$$ 4y = 16 $$

Divide both sides by 4:

$$ y = 4 $$

Now, substitute the value of $ y $ into the first equation to find $ x $:

$$ x = 3(4) - 4 $$

$$ x = 12 - 4 $$

$$ x = 8 $$

Therefore, the first number is $ x = 8 $.

Savannah Anderson · Answer

undefined Let's denote the first number as $ x $ and the second number as $ y $.

From the given information, we can set up two equations based on the problem statement:

1. The first number is 4 less than 3 times the second number:
$$ x = 3y - 4 $$

2. 3 more than two times the first number decreased by 2 times the second number equals 11:
$$ 2x + 3 - 2y = 11 $$

Now, we can substitute the expression for $ x $ from the first equation into the second equation:

$$ 2(3y - 4) + 3 - 2y = 11 $$

Simplify the equation:

$$ 6y - 8 + 3 - 2y = 11 $$
$$ 4y - 5 = 11 $$

Add 5 to both sides:

$$ 4y = 16 $$

Divide both sides by 4:

$$ y = 4 $$

Now, we can substitute the value of $ y $ into the first equation to find $ x $:

$$ x = 3(4) - 4 $$
$$ x = 12 - 4 $$
$$ x = 8 $$

So, the first number is 8.