A system of equations is given below:
y = 3x + 4
2x + 3y = 10
Which of the following steps could be used to solve by substitution? (4 points)
3x + 4 = 2x + 3y
2x + 3(3x + 4) = 10
2(3x + 4) + 3y = 10
3(y = 3x + 4)
(I don't understand this at all and I've read through the module multiple times. Could you please help?)
y = 3x + 4
2x + 3y = 10
Substitute 3x+4 for y in the second equation and solve for x. Insert that value into the first equation to solve for y. Check by putting both values into the second equation.
To solve a system of equations by substitution, you need to solve one of the equations for one variable, and then substitute that expression into the other equation. Here are the steps to solve the given system of equations by substitution:
1. Start with the given system of equations:
y = 3x + 4
2x + 3y = 10
2. Solve the first equation for y:
y = 3x + 4
3. Substitute the expression for y, which is 3x + 4, into the second equation:
2x + 3(3x + 4) = 10
4. Simplify the equation:
2x + 9x + 12 = 10
5. Combine like terms:
11x + 12 = 10
6. Subtract 12 from both sides of the equation:
11x = -2
7. Divide both sides of the equation by 11 to solve for x:
x = -2/11
8. Substitute the value of x back into the first equation to solve for y:
y = 3(-2/11) + 4
9. Simplify the equation:
y = -6/11 + 4
10. Combine like terms:
y = (-6 + 44)/11
11. Simplify the equation:
y = 38/11
So the solution to the system of equations is x = -2/11 and y = 38/11.
Now, let's go back to the options you provided:
1. 3x + 4 = 2x + 3y
This is not a correct step for solving the system by substitution because it doesn't involve substituting one equation into another.
2. 2x + 3(3x + 4) = 10
This is the correct step. It involves substituting the expression for y into the second equation of the system.
3. 2(3x + 4) + 3y = 10
This step is incorrect because it doesn't involve substituting the expression for y into the equation.
4. 3(y = 3x + 4)
This step is incorrect as well. It seems to be a typo, and the equal sign (=) should not be present within the brackets.
So, the correct step that could be used to solve the system by substitution is 2x + 3(3x + 4) = 10.
Of course! Solving a system of equations using the substitution method involves substituting one equation into the other in order to eliminate one variable. Here are the steps to solve this system of equations by substitution:
1. The given system of equations is:
y = 3x + 4
2x + 3y = 10
2. Choose one equation (either equation is fine) and solve it for one of the variables in terms of the other variable. Let's solve the first equation for y in terms of x:
y = 3x + 4
3. Substitute the expression for y (from step 2) into the other equation, replacing the y variable. We will substitute 3x + 4 for y in the second equation:
2x + 3(3x + 4) = 10
4. Simplify the equation obtained in step 3 by performing the multiplication:
2x + 9x + 12 = 10
5. Combine like terms on the left side of the equation:
11x + 12 = 10
6. Solve for x by isolating the variable. Subtract 12 from both sides of the equation:
11x = -2
7. Divide both sides of the equation by 11 to solve for x:
x = -2/11
8. Substitute the value of x (from step 7) into one of the original equations to solve for y. Let's use the first equation:
y = 3(-2/11) + 4
9. Simplify the equation obtained in step 8:
y = -6/11 + 4
10. Combine like terms on the right side of the equation:
y = (-6 + 44)/11
11. Simplify the equation:
y = 38/11
Therefore, the solution to the system of equations is x = -2/11 and y = 38/11.
Regarding the multiple-choice options you provided, none of them are correct because they do not represent the correct steps for solving the system of equations by substitution.