a Do either of the equations have an isolated variable?
B
b Do either of the equations have an isolated variable?
A
a (- 3, - 7)
B
b (5, - 1)
Practice makes perfect
a In this system of equations, at least one of the variables has a coefficient of 1. Therefore, we will approach its solution with the Substitution Method.
Solving Algebraically
When solving a system of equations using substitution, there are three steps.
Isolate a variable in one of the equations.
Substitute the expression for that variable into the other equation and solve.
Substitute this solution into one of the equations and solve for the value of the other variable. Since the expression equal to y in (I) is simpler, let's use that for our initial substitution.
The solution to this system of equations is the point (- 3, - 7).
Graphing
When graphing the given equations, the point of intersection indicates the solution to the system. To find it we will need the equations to be in slope-intercept form to identify the slope m and y-intercept b.
Let's rewrite each in slope-intercept form, highlighting the m and b values.
Given Equation
Slope-Intercept Form
Slope m
y-intercept b
y=4x+5
y=4x+ 5
4
(0, 5)
y=- 2x-13
y=-2x+( -13)
-2
(0, -13)
To graph these equations we will start by plotting their y-intercepts. Then we will use the slope to determine another point that satisfies each equation, and connect the points with a line.
We can see that the lines intersect at exactly one point.
The point of intersection at (-3,- 7) is the solution to the given system of equations. It confirms the answer, which we obtained algebraically.
b In this system of equations at least one of the variables has a coefficient of 1. Therefore, we will approach its solution with the Substitution Method.
Solving Algebraically
When solving a system of equations using substitution, there are three steps.
Isolate a variable in one of the equations.
Substitute the expression for that variable into the other equation and solve.
Substitute this solution into one of the equations and solve for the value of the other variable. For this exercise y is already isolated in one equation, so we can skip straight to solving!
The solution, or point of intersection, to this system of equations is the point (5, - 1).
Graphing
When graphing the given equations, the point of intersection indicates the solution to the system. To find it, we need the equations to be in slope-intercept form to identify the slope m and y-intercept b.
Let's rewrite each of the equations in slope-intercept form, highlighting the m and b values.
Given Equation
Slope-Intercept Form
Slope m
y-intercept b
2x+y=9
y=- 2x+ 9
- 2
(0, 9)
y=- x+4
y=- 1x+ 4
- 1
(0, 4)
To graph these equations we will start by plotting their y-intercepts. Then, we will use the slope to determine another point that satisfies each equation, and connect the points with a line.
We can see that the lines intersect at exactly one point.
The point of intersection at (5,- 1) is the solution to the given system of equations. It confirms the answer, which we obtained algebraically.
