Classwork 4-2 Worksheet 1

Examine the following system of linear equations. How many solutions will this system have? That is, how many (x_{1},y_{1}) pairs are there that will work to make all the equations in the system true at the same time? Explain your reasoning.

y=4x+2

y=-\frac{1}{2}x-1

Assume that the only solution to the system of equations is the ordered pair (\alpha, \beta). That is, each of the equations in the system is true when x=\alpha and y=\beta. Thus, we can write two new equations.

The first equation is \beta=4\alpha+2. Write the second, analogous equation.

Now combine these two equations to make a new, third equation. We will do this using the "transitive property of equality", which states:

Transitive property of equality:

Whenever a=b and b=c, then a=c.

For our purposes, you might prefer to think of this the way Euclid (the famous ancient Greek geometer) phrased it: "If two things are each equal to a third, then they are equal to each other."

Solve your equation from the question above.

If we have a linear equation, y=mx+b, and we know either the y-value or the x-value, we can find the other. Use this fact to continue from the findings above.

What ordered pair, (x,y) is a solution to the system of equations?

Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.

y=4x-3

y= -x+2

Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.

y= x-3

y=\frac{2}{5}x+3

Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.

y=x

y= -1x

Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.

y= -x+14

y=2x-22

Use substitution to find the solution to the following system of linear equations. Then check your answer graphically.

y= 22x+6

y= 44x+6

The following system of equations is written in standard form. As a first step to finding a solution to the system, convert each equation in the system to slope-intercept form.

2x-3y=7

-x+5y=2

Use the slope-intercept form equations to find the solution to the system.

The following system of equations includes linear equations in different forms. Rewrite ONLY THE FIRST of the equations in slope-intercept form.

y-2=3(x+1)

3x+y=9

Now that you have an equation which provides an equivalent expression for y, substitute that expression for y in the second equation from above. Write your resulting equation here.

What is the solution to the system of equations?