Example Card: Solving by Graphing
Solve systems of equations by graphing in Grade 9 algebra: graph each equation on the same coordinate plane and identify the intersection point as the solution to the system.
Key Concepts
Equations often come in disguise. Let's unveil their true linear form to find where they meet. This example card demonstrates the key idea of solving a system by graphing, including rearranging equations first.
Example Problem Solve the system by graphing and check your solution: $2x + y = 7$, $x y = 1$.
Step by Step 1. To solve by graphing, we first need to rewrite both equations into slope intercept form, $y = mx + b$. 2. Rearrange the first equation, $2x + y = 7$: $$ \begin{aligned} 2x + y &= 7 \\ \underline{ 2x \quad} & \quad \underline{ 2x} \\ y &= 2x + 7 \end{aligned} $$ This line has a y intercept of $7$ and a slope of $ 2$. 3. Rearrange the second equation, $x y = 1$: $$ \begin{aligned} x y &= 1 \\ \underline{ x \quad} & \quad \underline{ x} \\ y &= x 1 \\ y &= x + 1 \end{aligned} $$ This line has a y intercept of $1$ and a slope of $1$. 4. Graph both equations. The line $y = 2x + 7$ starts at $y=7$ and moves down 2 units for every 1 unit to the right. The line $y = x + 1$ starts at $y=1$ and moves up 1 unit for every 1 unit to the right. The lines intersect at the point $(2, 3)$. 5. Check that the ordered pair $(2, 3)$ makes both original equations true. $$ \begin{array}{c|c} 2x + y = 7 & x y = 1 \\ 2(2) + 3 \stackrel{?}{=} 7 & 2 3 \stackrel{?}{=} 1 \\ 4 + 3 \stackrel{?}{=} 7 & 1 = 1 \quad \checkmark \\ 7 = 7 \quad \checkmark \end{array} $$ 6. The ordered pair $(2, 3)$ is a solution to both equations, so it is the solution to the system.
Common Questions
How do you solve a system of equations by graphing?
Graph each equation on the same coordinate plane. The intersection point — where the two lines cross — is the solution of the system. If lines are parallel (no intersection), there is no solution; if they overlap, there are infinitely many solutions.
What does the intersection point represent in a graphed system?
The intersection point (x, y) gives the values that satisfy BOTH equations simultaneously. It is the only point that lies on both lines, making it the unique solution to the system.
When is solving by graphing less reliable than algebraic methods?
Graphing is approximate — the intersection may not land exactly on integer coordinates. For non-integer solutions like (1.5, 3.7), reading the exact values from a graph is difficult. Algebraic methods give exact solutions.