Find the values of x and y for the system: 7x - 15y = 2 and x + 2y = 3
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Step-by-Step Solution
Step 1: Express one variable in terms of the other
From the second equation, x+2y=3, we can easily express x in terms of y. This is a common first step in the substitution method, where we isolate one variable to substitute it into the other equation. We achieve this by subtracting 2y from both sides of the equation.
Step 2: Substitute the expression into the other equation
Now, we substitute the expression for x, which is 3−2y, into the first equation, 7x−15y=2. This substitution will result in a linear equation that contains only the variable y, making it solvable.
Step 3: Simplify and solve for y
We first distribute the 7 into the parenthesis, which gives us 21−14y. Then, we combine the like terms, −14y and −15y, to get −29y. Next, we subtract 21 from both sides of the equation and finally divide by −29 to find the value of y.
Step 4: Substitute the value of y back into the expression for x
Now that we have the value of y=2919, we substitute it back into the expression for x that we derived in the first step: x=3−2y. We multiply 2 by 2919 and then subtract the resulting fraction from 3, finding a common denominator to perform the subtraction.
Step 5: Verify the solution
To ensure our solution is correct, we substitute the calculated values of x=2949 and y=2919 into both original equations. If both equations hold true, our solution is verified. In this case, both equations are satisfied.