# KutaSoftware: Algebra 2- Systems Of Two Equations Part 1 (2023)

## Introduction

Free worksheet at www.kutasoftware.com/freeia2.html
Check out my graphing notebook: www.amazon.com/dp/B09TMYWFNG

Go to ➡️ maemap.com/math/algebra2/ ⬅️ for more Algebra 2 information!

Happy learning! 💜

## Content

In this video we're going to start the CUDA software, infinite algebra, 2, free worksheet systems of two equations, and in this section we're going to be solving each system by graphing and number one.

Each of our equations is in the form of y equals, MX plus B, so we'll use our slope and y-intercept in order to graph the lines so for y equals negative 3x plus 4.

Our slope is negative 3, which we can write as negative 3 over 1 for rise over run, and then our y-intercept B is a positive 4.

So we'll start by graphing, our y-intercept 0 4 and then we'll use our rise over run to find that next point so we'll go down: 3 1, 2, 3 and the rise and over 1 in the positive direction.

For the run to the point, 1 1 now we'll connect those points with a line.

So any point that falls on this line is a solution to our first equation.

However, we're solving the system so we're going to also need to graph our next equation for y equals 3x minus 2.

Our slope M is a positive 3, which we can write as 3 over 1 for rise over run and our y-intercept B is a negative 2.

So we'll graph plotting the point, 0 negative 2 for our y-intercept and then using our rise over run, we'll go up 3 and over to the right one to the point, 1 1.

So now that we have two points, we can connect those to form our next line.

Now.

Any point that falls on this line is a solution to that second equation.

However, in order to find a solution to both equations, we need to find a point that falls on both lines and therefore will be that intersection point, which is the point 1 1, so 1 1 is our solution and number 1, and you can double check by plugging 1 in for X in each equation and see if the right-hand sides equal 1 continuing on a number two.

Let's first graph y equals x, plus 2, our slope M is 1, so rise over run will be 1 over 1 and our y-intercept B is equal to a positive 2, so graphing 0 2 and then using our slope to get to that next point rise of 1 run of 1 will be at 1/3.

So then I'll draw a line connecting now it's time to graph that next line x equals negative 3.

So that's going to be a vertical line where X is always negative 3.

So if Y is 5, X is negative.

3 if Y is 1, X is negative 3 if Y is negative, 4 X is negative, 3 and so on.

Now the intersection of those two lines occurs at this point: negative 3, negative 1, so negative, 3 negative 1 is our solution in number 2, plugging negative 3 in for X negative 3 is indeed equal to negative 3 and negative 3 plus 2 does indeed equal a negative 1.

Now moving on to number 3 I want to write each of these in the form of y equals, MX plus B.

So that's easier to graph.

So 4x minus y equal to 3 I'll subtract X from both sides, so you get that negative y equals negative x plus 3.

However, I don't want my Y to be negative, so multiply both sides by a negative 1 or divide both sides by a negative 1, so I'll get that positive.

Y is equal to X minus 3, so my slope, M is 1 which I'll write as rise over run 1 over 1 and my y-intercept B is negative.

3 plotting my y-intercept 0 negative 3 and then using my slope, going up 1 in the Y and over to the right 1 and the X I'll, get to the point.

1 negative, 2 and all I need is two points to form a line.

Now it's time to grasp that next line, 7x minus y equals negative 3.

What I'll do is subtract 7x from both sides, so I'll have negative y equal to negative 7x.

Minus 3 and I'll need to divide both sides by negative 1 in order to get this Y positive, so I'll have positive y equal to positive, 7x, plus a positive 3.

So my slope, M is 7 which will write a 7 over 1 and my y-intercept B is a positive 3, so plotting our point, 0 3 and then using my rise over run.

I'll get to the next point, however, going positive 7 in the Y and positive 1 and the X will take me off the coordinate system.

So I'll write my rise over run as negative 7 over negative 1, because a negative, divided by a negative is still a positive.

Now I'll go down 7 and 1 to the left, to the point negative 1, negative, 4 and then I'll draw my line and where those lines intersect.

That is my answer and that occurs at negative, 1, negative 4, so negative, 1 negative 4 is my solution and number 3.

Lastly, in this video I'm going to go over number 4, however, before I go over the answer to this, don't forget to like this video and subscribe to my channel all likes and subscriptions are greatly appreciated, and they, let me know that you're finding these videos useful so go ahead, and do that now continuing on with number 4 we're going to write each equation in the form of y equals MX plus B.

So our first equation is 4x plus y equals 2, so I'll subtract 4x from both sides to get that y equals negative, 4x plus 2 for our second equation.

X minus y equals positive, 3 I'll subtract X from both sides to get that negative y equals negative, x, plus and then I'll divide or multiply both sides by negative one.

In order to flip this sign, on my left hand, side so I'll get positive y equals positive x minus three.

Now for my first line, my slope M is negative: four, which I'll write as negative four over one for rise over run and my y-intercept B is a positive 2, so plotting 0 2.

My y-intercept of my first line and then using my rise of a run I'll go down for M 1 to the right to get to that next point of 1, negative 2 and then I'll connect those with a line.

Next, looking at the slope of my other equation, M is equal to a positive 1, so rise over run 1 over 1 and my y-intercept B is a negative 3, so I'll plot, 0, negative 3 and then rise over run, moving one up and one to the right or moving one down and one to the left.

If I was to use negative 1 over negative 1, now I'll draw that line, and you can see that my equations intersect at the point 1 negative 2, so 1 negative 2 is my solution in number.

4 again, don't forget to click that like button and subscribe to my channel and share this with somebody that you think will also find it useful continue on to the next video, where we'll start by solving each system by substitution and before continuing on.

If you have any questions, leave them in the comments below.

## FAQs

### How do you find the answer of two equations? ›

How do I solve systems of linear equations by substitution?
1. Isolate one of the two variables in one of the equations.
2. Substitute the expression that is equal to the isolated variable from Step 1 into the other equation. ...
3. Solve the linear equation for the remaining variable.

What is the solution to the system of equations answers? ›

The solution set to a system of equations will be the coordinates of the ordered pair(s) that satisfy all equations in the system. In other words, those values of x and y will make the equations true. Accordingly, when a system of equations is graphed, the solution will be all points of intersection of the graphs.

How many solutions to a system of equations worksheet? ›

There are THREE TYPES OF SOLUTIONS to any system of equations: 1) ONE SOLUTION: Exactly one ordered pair (x, y) will solve all equations. 2) NO SOLUTION: NO ordered pair (x, y) will EVER solve all equations. 3) INFINITELY MANY SOLUTIONS: an infinite number of ordered pairs will solve all equations.

How do you solve a system of equations step by step? ›

Key Concepts
1. Write both equations in standard form. ...
2. Make the coefficients of one variable opposites. ...
3. Add the equations resulting from Step 2 to eliminate one variable.
4. Solve for the remaining variable.
5. Substitute the solution from Step 4 into one of the original equations. ...
6. Write the solution as an ordered pair.
Jan 5, 2020

What is the easiest way to solve system of equations? ›

HOW TO SOLVE A SYSTEM OF EQUATIONS BY ELIMINATION.
1. Write both equations in standard form. ...
2. Make the coefficients of one variable opposites. ...
3. Add the equations resulting from Step 2 to eliminate one variable.
4. Solve for the remaining variable.
5. Substitute the solution from Step 4 into one of the original equations.
May 3, 2019

What is the solution to the system of equations quiz? ›

The solution to a system of equations is any ordered pair that makes both equations true.

How do you find all solutions to a system of equations? ›

To solve a system of equations by graphing, graph all the equations in the system. The point(s) at which all the lines intersect are the solutions to the system. Graph of System Since the two lines intersect at the point (1, 1), this point is a solution to the system.

How many solutions does a system of two equations have? ›

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line).

How many solutions can a system of 2 equations have? ›

Under normal circumstances a system of two linear equations can have 0 , 1 or infinitely many solutions.

What are all the formulas for Algebra 2? ›

Important Formulas in Algebra
• a2 – b2 = (a – b)(a + b)
• (a + b)2 = a2 + 2ab + b2
• a2 + b2 = (a + b)2 – 2ab.
• (a – b)2 = a2 – 2ab + b2
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.
• (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca.
• (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)

### How many equations do you need to solve for 2 variables? ›

If the goal is to find x and y intersection, one question can't answer that. You must have at least 2 equations to solve the two variables, since you can't have an intersection point w one equation.

How many solutions can a system of 2 linear equations in 2 variables have? ›

How many solutions are there for linear equations in two variables? For linear equations in two variables, there are infinitely many solutions.

How do you solve a multi step one step two step equation? ›

Solving Multi-Step Equations
1. (Optional) Multiply to clear any fractions or decimals.
2. Simplify each side by clearing parentheses and combining like terms.
3. Add or subtract to isolate the variable term—you may have to move a term with the variable.
4. Multiply or divide to isolate the variable.
5. Check the solution.

What are the 4 steps to solving an equation? ›

We have 4 ways of solving one-step equations: Adding, Substracting, multiplication and division.

Which is the most efficient method to solve the system of equations? ›

If both equations are given in standard form ( A x + B y = C ) , then linear combinations is usually most efficient.

What are the methods of solving algebraic equations? ›

The algebraic method is a collection of several methods used to solve a pair of linear equations with two variables. The most-commonly used algebraic methods include the substitution method, the elimination method, and the graphing method.

How do you solve a system of equations without graphing? ›

To solve a system of linear equations without graphing, you can use the substitution method. This method works by solving one of the linear equations for one of the variables, then substituting this value for the same variable in the other linear equation and solving for the other variable.

Can a system of equations have two answers? ›

No, a system of linear equations cannot have exactly two solutions.

What kind of lines have no solution to a system of equations? ›

If the system consists of two functions, then there will be no points of intersection. A system of two linear equations has no solution if the lines are parallel. Parallel lines on a coordinate plane have the same slope and different y-intercepts (see figure 3 for an example of this).

Which method would be the simplest way to solve the system 7x 5y 19 7x 2y =- 16? ›

Elimination. Distributive. Hence, the elimination method is the simplest way to solve the given equations.

### How many solutions does this linear system have y =- 6x 2 12x 2y =- 4? ›

The linear system of equations y = -6x + 2 and -12x - 2y = -4 has infinitely many solutions.

How many solutions does the equation || 2x 3 |- M |= m have if M 0? ›

Therefore, the equation ||2x-3|-m|=m has two solutions for any value of m > 0.

How do you tell if a system of equations has no solution or infinitely many? ›

If you get a unique solution for each variable, there is one solution. If you get a contradiction like 0 = 1, then there is no solution. If you get an equation that is always true, such as 0 = 0, then there are infinite solutions.

How many solutions do two lines or two equations in a system have if they have the same slope and same Y-intercept? ›

These two equations have the same slope and the same y-intercept. These equations create one line. This system is an inconsistent system because it has an infinite number of solutions.

What is system of two equations with infinitely many solutions? ›

The system of an equation has infinitely many solutions when the lines are coincident, and they have the same y-intercept. If the two lines have the same y-intercept and the slope, they are actually in the same exact line.

How many solutions will there by if two equations are the same line? ›

If the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations. If the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations.

How hard is algebra 2? ›

In addition, Algebra 2 is the first math class in a student's math career that introduces topics that are more complex and less concrete, like complex numbers or logarithms, which makes Algebra 2 harder to grasp than other math classes whose concepts are more straight forward and easier to visualize.

What grade level is algebra 2? ›

Students typically learn Algebra II in 11th grade. An Algebra II curriculum usually builds on knowledge and skills that are gained in Algebra I and reinforced in Geometry, including relationships between quantities through equations and inequalities, graphing of functions, and trigonometry.

Is algebra 2 basic math? ›

Algebra 2 is the advanced level of pre-algebra and Algebra 1. It introduces higher grades topics such as evaluating equations and inequalities, matrices, vectors, functions, quadratic equations, complex numbers, relations, inverse operations, and various other properties.

What is the new 2x2 system? ›

When a system has two equations and two variables, it is a 2x2 system of equations. A solution to a system of equations consists of values of the variables in the system that makes each of the equations true.

### How do you complete a solution to two variable equations? ›

A two-variable equation is solved by plugging in x = 0 and solving for y and then plugging in y = 0 and solving for x. These results give you the x and y intercepts of the graph. By joining those two points, the equation will be represented.

Is it possible for 2 equations to have more than 1 solution? ›

There can be more than one solution to a system of equations. A system of linear equations will have one point of intersection, or one solution. To graph a system of equations that are written in standard form, you must rewrite the equations in slope -intercept form.

Which equation has infinitely many solutions? ›

An infinite solution has both sides equal. For example, 6x + 2y - 8 = 12x +4y - 16. If you simplify the equation using an infinite solutions formula or method, you'll get both sides equal, hence, it is an infinite solution.

Can an equation have 2 answers? ›

A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.

What is it called when 2 equations have the same answer? ›

Systems of equations that have the same solution are called equivalent systems. Given a system of two equations, we can produce an equivalent system by replacing one equation by the sum of the two equations, or by replacing an equation by a multiple of itself.

How do I find an equation of 2 lines from a pair of straight lines? ›

Ans. If a pair of straight lines is represented by the equation ax2+2hxy+by2+2gx+2fy+c=0, then the equation m1+m2=–2h/b, and m1m2=a/b, where m1 and m2 are the slopes of the straight lines. We may easily determine the slope of a pair of straight lines using the relations that have been provided.

What is an example of two equations? ›

For example, 10x+4y = 3 and -x+5y = 2 are linear equations in two variables. The solution for such an equation is a pair of values, one for x and one for y which further makes the two sides of an equation equal.

Is the answer 16 or 1? ›

Experts say that depending on where in the world you learned math determines how you can solve the problem, there is the PEMDAS method and the BODMAS method. With that in mind, the answer is 1.

What is it called when two algebraic expressions set equal to each other? ›

An equation is a mathematical statement that two expressions are equal.

What is a system of equations with no solution called? ›

If a system has no solution, it is said to be inconsistent .

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