Solve the equation 5x-3y=14 and 3x+5y=18 using substitution.
The ans is x=3 11/17 and y= 1 7/17
I couldn't get the above answers, could someone show me clearly how to get those answers, thank you.
Bad answer
To solve the system of equations using substitution, we can solve one equation for one variable and substitute it into the other equation. Let's solve the first equation for x:
1) 5x - 3y = 14
First, isolate x by moving the -3y term to the other side of the equation:
5x = 14 + 3y
Next, divide both sides of the equation by 5 to solve for x:
x = (14 + 3y) / 5
Now, substitute this expression for x into the second equation:
2) 3x + 5y = 18
Replace x with (14 + 3y) / 5:
3((14 + 3y) / 5) + 5y = 18
Multiply both sides of the equation by 5 to eliminate the fraction:
3(14 + 3y) + 25y = 90
Distribute the 3 to both terms inside the parentheses:
42 + 9y + 25y = 90
Combine like terms:
34y + 42 = 90
Subtract 42 from both sides of the equation:
34y = 48
Divide both sides of the equation by 34 to solve for y:
y = 48 / 34
Simplify the fraction:
y = 24 / 17
Now, substitute this value of y back into the expression for x:
x = (14 + 3(24 / 17)) / 5
Multiply 3 by 24/17:
x = (14 + 72/17) / 5
Convert 14 to have a denominator of 17:
x = (238/17 + 72/17) / 5
Combine the two fractions:
x = 310/17 / 5
Divide 310/17 by 5:
x = 310/85
To simplify the fraction, we can express it as a mixed number:
Divide 310 by 85:
310 ÷ 85 = 3 with a remainder of 55
Therefore, x = 3 55/85, which can be simplified to 3 11/17.
So, the solutions to the system of equations using substitution are x = 3 11/17 and y = 24/17, or as a mixed number y = 1 7/17.
To solve the system of equations using substitution, we can solve one equation for one variable and substitute it into the other equation. Let's start by solving the first equation for x:
1) 5x - 3y = 14
Rearrange the equation to solve for x:
5x = 3y + 14
Divide both sides by 5:
x = (3y + 14)/5
Now, substitute this expression for x into the second equation:
2) 3x + 5y = 18
Replace x with (3y + 14)/5 in equation 2:
3((3y + 14)/5) + 5y = 18
Multiply through by 5 to eliminate the fraction:
3(3y + 14) + 25y = 90
Expand and simplify:
9y + 42 + 25y = 90
Combine like terms:
34y + 42 = 90
Subtract 42 from both sides:
34y = 48
Divide both sides by 34:
y = 48/34 = 24/17
Now that we have the value of y, we can substitute it back into equation 1 to find x:
5x - 3(24/17) = 14
Multiply 3 by 24/17:
5x - (72/17) = 14
Move (72/17) to the other side of the equation:
5x = 14 + (72/17)
To add 14 and (72/17), we need a common denominator:
5x = (238/17) + (72/17)
Combine fractions with the same denominator:
5x = (238 + 72) / 17
Simplify:
5x = 310 / 17
Divide both sides by 5:
x = (310 / 17) / 5
To divide by a fraction, we multiply by its reciprocal:
x = (310 / 17) * (1/5)
Multiply the numerators and the denominators:
x = 310 / (17 * 5)
Simplify:
x = 310 / 85
The final answer is x = 3 11/17 and y = 1 7/17.
From the first equation, x = (3y+14)/5
Use that in the other equation to get
3(3y+14)/5 + 5y = 18
9y+42 + 25y = 90
34y = 48
y = 48/34 = 24/17 = 1 7/17
x = (3y+14/5) = (72/17 + 14)/5 = 310/85 = 3 55/85 = 3 11/17
You could also have used other substitutions for x, or using the other equation to start with.