Welcome to SAT Math substitution problems. The substitution method is one of the most important techniques for solving systems of equations. We start with two or more equations and systematically eliminate variables by substitution. Let's explore the step-by-step process using a concrete example with two equations and two unknowns.
Now let's work through our example step by step. We have the system: 2x plus y equals 7, and x minus y equals 2. For step 1, we need to solve one equation for one variable. Looking at our equations, the second equation x minus y equals 2 is easier to work with. Let's solve it for x by adding y to both sides. This gives us x equals 2 plus y. Perfect! Now we have x expressed in terms of y.
Now for step 2, we substitute our expression x equals 2 plus y into the first equation. The first equation was 2x plus y equals 7. Substituting, we get 2 times the quantity 2 plus y, plus y equals 7. Let's expand this: 2 times 2 plus y gives us 4 plus 2y. So we have 4 plus 2y plus y equals 7. Combining like terms, we get 4 plus 3y equals 7. Subtracting 4 from both sides gives us 3y equals 3. Finally, dividing by 3, we find that y equals 1.
Now for step 3, we need to find the value of x. We know from step 2 that y equals 1. From step 1, we had the expression x equals 2 plus y. Let's substitute y equals 1 into this expression. We get x equals 2 plus 1, which simplifies to x equals 3. So our complete solution is x equals 3 and y equals 1. This gives us the point where the two lines intersect.
Finally, step 4 is to check our solution. We found x equals 3 and y equals 1. Let's substitute these values back into both original equations. For equation 1, 2x plus y equals 7: substituting gives us 2 times 3 plus 1, which equals 6 plus 1, equals 7. Check! For equation 2, x minus y equals 2: substituting gives us 3 minus 1, which equals 2. Check! Both equations are satisfied, so our solution is correct. The final answer is the ordered pair 3 comma 1.