Solve the system of linear equations by substitution:
1.) y=2x+3
y=3x+5
2.) y=1/3x+2
y=1/6x+4
just substitute the first value for y into the second equation:
2x+3 = 3x+5
Now solve for x, and y follows.
Same for the other one.
To solve a system of linear equations by substitution, follow these steps:
1. Obtain one of the equations and solve for one variable (either x or y) in terms of the other variable.
2. Substitute the expression obtained in step 1 into the other equation.
3. Solve the resulting equation for the remaining variable.
4. Substitute the value obtained in step 3 back into the expression found in step 1 to solve for the other variable.
Now let's solve the given systems of linear equations:
1.) y = 2x + 3 and y = 3x + 5
Step 1: Solve one equation for one variable
Let's solve the first equation, y = 2x + 3, for y.
From this equation, we can express y in terms of x: y = 2x + 3.
Step 2: Substitute the expression into the other equation
Substitute the expression for y from step 1 into the second equation, y = 3x + 5.
We have (2x + 3) = 3x + 5.
Step 3: Solve the resulting equation for x
Now, we just need to solve the equation (2x + 3) = 3x + 5 for x by isolating the x term.
Subtract 2x and 5 from both sides of the equation:
2x - 3x = 5 - 3
-x = 2
Step 4: Substitute the value back into the expression to solve for y
Now that we have x = -2, substitute it into y = 2x + 3 to find y.
y = 2(-2) + 3
y = -4 + 3
y = -1
So the solution to the system of equations is x = -2 and y = -1.
2.) y = (1/3)x + 2 and y = (1/6)x + 4
Let's solve this system using the same steps:
Step 1: Solve one equation for one variable
Let's solve the first equation, y = (1/3)x + 2, for y.
From this equation, we can express y in terms of x: y = (1/3)x + 2.
Step 2: Substitute the expression into the other equation
Substitute the expression for y from step 1 into the second equation, y = (1/6)x + 4.
We now have (1/3)x + 2 = (1/6)x + 4.
Step 3: Solve the resulting equation for x
To get rid of the fractions, we can multiply the equation by a common denominator, which is 6.
Multiply both sides of the equation by 6:
6[(1/3)x + 2] = 6[(1/6)x + 4]
2x + 12 = x + 24
Subtract x from both sides to isolate x:
2x - x + 12 = 24
x + 12 = 24
Subtract 12 from both sides:
x = 24 - 12
x = 12
Step 4: Substitute the value back into the expression to solve for y
Now that we have x = 12, substitute it into y = (1/3)x + 2 to find y.
y = (1/3)(12) + 2
y = 4 + 2
y = 6
The solution to the system of equations is x = 12 and y = 6.