Algebra 1 : How to find the solution for a system of equations

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #31 : How To Find The Solution For A System Of Equations

Find the solution for the system of equations.

 and 

Possible Answers:

 and 

 and 

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

A system of equations can be solved by subsituting one variable for another. Since we know that , we can subsitute this into the other equation so . This expression can be solved to find that . Now that we know the value of  it can be subsituted into either of the original equations to find .

Example Question #31 : How To Find The Solution For A System Of Equations

Possible Answers:

Correct answer:

Explanation:

Example Question #61 : Equations / Solution Sets

Determine the intersection point of the following two equations.

Possible Answers:

Correct answer:

Explanation:

To find the intersection point, you must find the values of x and y that satisfy the two equations. We can use the method of adding the two equations together:

If we add the equations together, the y terms cancel out, so we get

Now that we know the value of x, we can plug that into one of the equations and solve for y. Pluggin it into the first equation, we get

.

So that point of intersection is , or .

 

Example Question #31 : How To Find The Solution For A System Of Equations

Find the solution(s), if there is one, to this system of equations:  and 

Possible Answers:

Correct answer:

Explanation:

There are several methods to find the solution set to this system of equations, but here we will use the substitution method for its ease of application. The idea of this method is to solve one of the equations (and you can choose either) in terms of one variable and then plug it into the second equation that was not tampered with to solve it for the other variable. Do not worry if this isn't completely clear just yet. But after you read the step-by-step solution review this paragraph and make sure the math language and this explanation align in your mind.

1) 

2) 

First solve equation 2 for y. Again, it is your choice which equation and which variable to use, but try and select one that will take the least work. In our case neither are "better", but the bottom equation will not yield a fraction.

move over x term

isolate y

Now we have equation 2 in terms of one variable (y=....). We must now plug this equation we just made back into one of the original equations to solve that equation for one variable. Let's do equation 1:

Notice how all we've done is replace the y in equation 1 with the modified version of equation 2. Now solve for x:

Simplify the parenthesis:

Combine x's and move the constant over:

Now we have solved for one variable and are almost done! Plug this value into either original equation and solve for y. Let's plug it into equation 2:

move the constant over after simplifying:

Final answer= 

We now have solved for each variable. Thus our solution set to this system of equations, where they are equal to each other, or where their lines intersect is (2.2,-0.4)! Since there are multiple ways to solve this problem, there are mutiple ways to check yourself. Convert both equations to slope intercept form, graph them using a graphing utility, and use the trace or intersect function to see that these two lines really do intersect and therefore equal each other at this coordinate point. Alternately just plug in the coordinate pair to either ORIGINAL equation, but the graphical method is probably easy since we have decimals.

Example Question #31 : How To Find The Solution For A System Of Equations

Solve the system of linear equations for and .

 

Possible Answers:

Correct answer:

Explanation:

The question is asking the student to solve the linear set of equations ultimately by isolating and .

There are a few ways a student could choose to answer this. 

One may see immediately  and realize to eliminate the result from the second equation one would subtract as follows.

--

Other students may choose to subtract the first equation twice from the second equation then subsequently solve for y.

Example Question #62 : Equations / Solution Sets

Find where these two equations intersect:

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

The point these two equations intersect is their solution. There are several ways to solve and one should choose the method that makes the problem easiest. Here one can easily eliminate the two 4x variables. Thus, this problem will be solved by elimination.

1) 

2) 

Multiply equation 1 by -1:

Add equation 1 to equation 2:

1a)  

2)    

----------------------

Now substitute for y in either of the original equations:

2) 

   

 

 

Example Question #2437 : Algebra 1

Find the solution set to this system of equations:

Possible Answers:

Correct answer:

Explanation:

The solution set to this system of equations is a coordinate point whose x and y values would satisfy the two equations. This is the same place the two equations intersect. The elimination method will be used to solve for the solution set.

1) 

2) 

Multiply equation 1 by -2:

Add the new equation 1 to equation 2:

--------------------------

Substitute this x value back into either original equation to solve for y:

Example Question #2438 : Algebra 1

Which answer shows the solution to the system of equations?

Possible Answers:

Correct answer:

Explanation:

First step is to solve one of the equations for one of the variables.

Choose the equation easiest to solve for one of the variables.

.             

Substitute  for  in the other equation and solve for .

Use distributive property.

                         

Substitute  for  in either equation and solve for .

              

The solution is 

Example Question #71 : Equations / Solution Sets

Use elimination to solve the solution:

Possible Answers:

Correct answer:

Explanation:

For elimination you need to get one variable by itself by cancelling the other out. In this equation this is best done by getting rid of . You can multiply whichever equation you would like to, but multiply it by  to get

 

then add the equations together

which, simplified, is

divied by  to get  

Then plug  back into any equation for the x value

  

Solve for  to get

Example Question #72 : Equations / Solution Sets

In the standard coordinate plane, slope-intercept form is defined for a straight line as , where  is the slope and  is the point on the line where .

Give the coordinates at which the following lines intersect:

Possible Answers:

Correct answer:

Explanation:

The first step is to set the two equations equal to each other, since at the point of intersection, they will be. To do this more easily, convert each equation into slope-intercept form.

First equation:

 State equation

 Add  to both sides.

Second equation:

 State equation

 Divide both sides by .

 Subtract  from both sides.

Now, since each equation equals , the equations also equal each other (for the point of intersection). By solving for , therefore, we can 

 State equation.

 Subtract  from both sides.

 Subtract  from both sides.

 Divide both sides by .

 

So, the -coordinate of our intersection is . To find the -coordinate, plug this result back into one of the original equations.

 State your chosen equation.

 Substitute the value of .

So, the coordinates where the two lines intersect are .

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