This blog post provides comprehensive solutions to a Grade 10 math test focused on linear systems, covering methods such as graphing, substitution, and elimination, along with application problems involving real-world scenarios.
In this blog post, we will explore the solutions to the first Grade 10 math test, which focuses on solving linear systems. We will cover various methods including graphing, substitution, and elimination, and we will also tackle some application questions that demonstrate how these concepts can be applied in real-world scenarios.
The first question presents a linear system made up of two lines. To solve this system by graphing, we need to find the point where the two lines intersect. This involves rewriting both equations in the slope-intercept form (y = mx + b).
Equation 1: Start with the equation and isolate y:
Equation 2: This equation is already in slope-intercept form:
Next, we plot both lines on a graph:
Upon graphing, we find that the lines intersect at the point (0, -3). Thus, the solution to this system is:
For the second part, we again rewrite the equations in slope-intercept form:
Now, we graph both lines:
The lines intersect at (-3, 1), giving us the solution:
To solve the linear system using substitution, we start by isolating a variable:
Now, substitute x back into Equation 1 to find y:
Thus, the solution is:
For the second part, we isolate y in Equation 1:
Substituting into Equation 2:
The solution is:
To solve using elimination, we write the equations on top of each other and look for coefficients:
Adding these equations eliminates y:
Substituting back gives y = 8, so:
For the second part, we need to adjust the coefficients:
The solution is:
George invested $22,000 in two accounts. Let x be the amount in the account earning 4.5% and y be the amount in the account losing 2%.
Solving these gives:
Martha mixes cream. Let x be the amount of 10% cream and y be the amount of 2% cream.
Solving gives:
Let x be the speed of the canoe in still water and y be the speed of the current.
Solving gives:
Let x be the number of quarters and y be the number of dimes.
Solving gives:
In this post, we have covered various methods for solving linear systems, including graphing, substitution, and elimination. We also explored real-world applications of these methods through investment, mixing, and distance problems. Mastering these techniques is essential for success in Grade 10 math and beyond.
