Question 95·Medium·Systems of Two Linear Equations in Two Variables
The ordered pair satisfies the system of equations shown. What is the value of ?
(Express the answer as an integer)
For systems of two linear equations, quickly scan the coefficients to see if elimination is easy. If one variable has coefficients that are already opposites (like and here), add the equations to eliminate that variable in one step. Then solve the resulting one-variable equation. Since the question only asks for , you do not need to solve for , which saves time. Double-check your arithmetic when combining equations and when dividing to isolate the variable.
Hints
Choose a variable to eliminate
Look at the coefficients of and in both equations. Is there a variable whose coefficients make it easy to cancel out by adding or subtracting the equations?
Use elimination
Since one equation has and the other has , think about what happens if you add the two equations together. What happens to the -terms?
Solve the resulting equation
After you add the equations and the -terms disappear, you will get an equation that has only . Solve that one-step equation to find .
Desmos Guide
Enter the equations
In the first Desmos line, type 4x+5y=18. In the second line, type 6x-5y=12. Desmos will graph both lines.
Find the intersection point
On the graph, click or tap where the two lines intersect. Desmos will display the coordinates of this point as .
Read the value of x
Look at the x-coordinate of the intersection point. That x-value is the solution for in the system and is the number you should enter as your answer.
Step-by-step Explanation
Eliminate one variable by adding the equations
Notice that the coefficients of are and , which are opposites. This makes elimination by addition easy.
Add the two equations side by side:
The and cancel out:
Now you have a simple equation with only .
Solve for x
Solve the equation by isolating .
Divide both sides by :
So, the value of is .